#6915794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fg,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308682, #72308931, #72309031
maximal normal subgroup of	#72308682, #72308931, #72309031

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6822990);