#72287257

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	C₄²
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2662 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287278, #72287390, #72320746
maximal normal subgroup of	#72287278, #72287390, #72320746

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, 4, 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, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);