#7564236

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1725 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)(13,14,20,26)(15,21,29,23)(16,30,32,41)(17,35,28,37)(18,25,19,24)(22,34,40,33)(27,38,42,44)(31,36,43,39);(1,2)(3,4)(6,10)(11,12)(13,15,20,29)(14,21,26,23)(16,31,32,43)(17,18,28,19)(22,38,40,44)(24,35,25,37)(27,34,42,33)(30,36,41,39);(5,7,9,8)(6,11,10,12)(13,16,18,33,20,32,19,34)(14,22,24,30,26,40,25,41)(15,27,17,31,29,42,28,43)(21,36,37,38,23,39,35,44);(2,4)(13,17,19,15,20,28,18,29)(14,23,25,37,26,21,24,35)(16,31,34,27,32,43,33,42)(22,39,41,38,40,36,30,44);(5,8,9,7)(6,11,10,12)(16,32)(22,40)(27,42)(30,41)(31,43)(33,34)(36,39)(38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,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, 7471432);