#7564240

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1750 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)(13,14,20,25)(15,21,28,38)(16,29,32,41)(17,35,27,37)(18,24,19,23)(22,34,40,33)(26,39,42,44)(30,36,31,43);(1,2)(3,4)(6,10)(11,12)(13,15,20,28)(14,21,25,38)(16,30,32,31)(17,19,27,18)(22,39,40,44)(23,37,24,35)(26,34,42,33)(29,36,41,43);(5,7,9,8)(6,11,10,12)(13,16,19,34,20,32,18,33)(14,22,24,41,25,40,23,29)(15,26,17,31,28,42,27,30)(21,36,37,44,38,43,35,39);(2,4)(13,17,19,28,20,27,18,15)(14,21,24,37,25,38,23,35)(16,31,34,42,32,30,33,26)(22,36,41,44,40,43,29,39);(5,8,9,7)(6,11,10,12)(16,32)(22,40)(26,42)(29,41)(30,31)(33,34)(36,43)(39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=gi,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^c=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, 7471436);