#72287637

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	C₂²×#41	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,6)(7,12)(8,13)(14,16)(18,24)(25,38)(26,31)(27,39)(28,40)(41,45)(42,48)(43,44);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18)(19,25)(20,31)(21,27)(22,28)(23,24)(26,34)(29,45)(30,38)(32,48)(33,44)(35,43)(36,39)(37,40)(41,47)(42,46);(1,3,4,10,5,9,11,15)(2,7,8,14,6,12,13,16)(17,19,21,22,23,30,36,37)(18,25,27,28,24,38,39,40)(20,32,33,29,34,46,35,47)(26,42,43,41,31,48,44,45);(2,6)(3,9)(8,13)(10,15)(18,24)(25,38)(26,31)(27,39)(28,40)(41,45)(42,48)(43,44);(17,20,22,32,21,35,30,47,23,34,37,46,36,33,19,29)(18,26,28,42,27,44,38,45,24,31,40,48,39,43,25,41)

Polycyclic presentation:

2^10|c^2=f,e^2=g,f^2=h,g^2=i,i^2=j,b^a=bh,d^b=dh,d^c=dhj,e^b=ej,e^c=ei,f^e=fj,g^c=gj

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287684, #72287790, #72318539, …
maximal normal subgroup of	#72287684, #72287790, #72318539, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);