#7613575

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

2007 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)(7,10)(8,9)(11,12)(13,14)(15,21)(16,30)(17,32)(18,25)(19,24)(20,26)(22,29)(23,35)(27,43)(28,38)(31,37)(33,39)(34,42)(36,41)(40,44);(2,4)(6,9)(10,12)(13,15,20,29)(14,21,26,22)(16,18,28,19)(17,33,34,31)(23,40,41,43)(24,30,25,38)(27,35,44,36)(32,39,42,37);(5,7,8,11)(6,10,9,12)(13,16,18,29,20,28,19,15)(14,22,24,30,26,21,25,38)(17,31,35,44,34,33,36,27)(23,39,32,40,41,37,42,43);(1,2,3,4)(13,17,19,36,20,34,18,35)(14,23,25,42,26,41,24,32)(15,27,28,33,29,44,16,31)(21,37,30,40,22,39,38,43);(6,9)(10,12)(17,34)(23,41)(27,44)(31,33)(32,42)(35,36)(37,39)(40,43)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=h,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, 7520771);