#7604237

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=hi,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7511433);