#7519258

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

2610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=i,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,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426454);