#7168331

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994171
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1271 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7075527);