#7614420

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#45269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1844 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)(7,11)(8,12)(9,10)(13,14,19,24)(15,25,29,36)(16,30,33,38)(17,23,18,22)(20,28,37,26)(21,31,40,34)(27,39,32,35)(41,43,44,42);(2,4)(6,10)(11,12)(15,26)(16,31)(17,18)(20,25)(21,38)(22,23)(28,29)(30,40)(33,34)(36,37)(41,44)(42,43);(5,7,9,8)(6,11,10,12)(13,15,17,26,19,29,18,28)(14,20,22,25,24,37,23,36)(16,32,34,43,33,27,31,42)(21,39,30,44,40,35,38,41);(1,2,3,4)(13,16,18,31,19,33,17,34)(14,21,23,38,24,40,22,30)(15,27,28,43,29,32,26,42)(20,35,36,44,37,39,25,41);(5,8,9,7)(6,11,10,12)(16,33)(21,40)(27,32)(30,38)(31,34)(35,39)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=h,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,d^c=di,e^d=ei

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, 7521616);