#7502886

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 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)(13,14,18,23)(15,24,19,20)(16,27,29,38)(17,30,31,34)(21,32,36,25)(22,26,37,33)(28,41,42,35)(39,40,43,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,19)(14,20,23,24)(16,25,29,32)(17,26,31,33)(21,27,36,38)(22,34,37,30)(28,39,42,43)(35,40,41,44);(2,4)(13,16,18,29)(14,21,23,36)(15,25,19,32)(17,28,31,42)(20,27,24,38)(22,35,37,41)(26,39,33,43)(30,44,34,40);(6,11)(7,12)(13,17,18,31)(14,22,23,37)(15,26,19,33)(16,28,29,42)(20,34,24,30)(21,35,36,41)(25,39,32,43)(27,40,38,44);(5,8,9,10)(6,7,11,12)(17,31)(22,37)(26,33)(28,42)(30,34)(35,41)(39,43)(40,44)

Polycyclic presentation:

2^9|a^2=i,b^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=dfi,g^c=gi,e^d=ei,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, 7410082);