#7503714

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^c=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, 7410910);