#6865744

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1663 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,27)(20,22)(24,31)(28,32)(29,30);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,18,21,23)(19,28,29,24)(20,26,22,25)(27,31,30,32);(11,16)(13,15)(17,19,21,29)(18,24,23,28)(20,27,22,30)(25,31,26,32);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,20,21,22)(18,25,23,26)(19,27,29,30)(24,31,28,32);(1,5,6,4)(2,7,3,8)(17,20,21,22)(18,26,23,25)(19,27,29,30)(24,32,28,31)

Polycyclic presentation:

2^9|b^2=f,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,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340634, #72340639, #72340653
maximal normal subgroup of	#72340634, #72340639, #72340653

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6772940);