#6461113

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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
prime radical	#35226	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1761 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)(9,10)(11,13)(12,14)(15,16)(18,22)(19,25)(21,27)(23,30)(26,31)(28,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,26)(20,22)(21,24)(23,28)(25,30)(27,29)(31,32);(10,14)(11,15)(17,19,27,32,20,28,21,25)(18,23,29,30,22,26,24,31);(11,15)(13,16)(19,28)(23,26)(25,32)(30,31);(1,3,4,7)(2,6,5,8)(18,22)(23,26)(24,29)(30,31)

Polycyclic presentation:

2^9|c^2=gh,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368309);