#6865343

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667652
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×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251211, #72251222, #72251266, …
maximal normal subgroup of	#72251211, #72251222, #72251266, …
maximal quotient of	#72308858, #72308932

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6772539);