#7613821

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1901 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7521017);