#7186862

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996315
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fg,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh

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, 7094058);