#7502718

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1836 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409914);