#6631809

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631112
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 ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1736 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)(3,5)(4,6)(7,8)(9,10,13,16,12,14,11,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,3,4,8)(2,5,6,7)(10,14)(15,16)(18,22)(23,28)(24,29)(30,32);(1,2)(3,7)(4,6)(5,8)(10,15)(11,13)(14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(2,6)(5,7)(9,11,12,13)(10,15,14,16)(19,25)(23,28)(26,31)(30,32);(10,14)(15,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,c^b=ch,d^c=dfh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308682, #72309000
maximal normal subgroup of	#72308682, #72309000

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6539005);