#72328023

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁶
commutator	C₂³	C₂⁵×C₄
prime radical	C₂²×#3484	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

55197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(2,4)(5,6,7,8)(11,16)(13,14)(17,18)(19,22)(20,23)(21,24);(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23);(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24);(2,4)(5,6,7,8)(11,16)(13,14);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,23)(22,24);(9,10)(11,13)(12,15)(14,16)(17,19)(18,22)(20,24)(21,23)

Polycyclic presentation:

2^10|a^2=g,d^2=g,b^a=bgh,c^a=cj,c^b=cghj,d^c=dj,e^a=egi,e^d=egi,f^e=fghj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328045, #72332461
maximal normal subgroup of	#72328045, #72332461
maximal quotient of	#72333246, #72335832

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