#6865528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257431, #72257512, #72264068
maximal normal subgroup of	#72257431, #72257512, #72264068
maximal quotient of	#72308915

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6772724);