#7564566

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1934 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=g,e^2=fi,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^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7471762);