#6479638

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
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×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1742 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)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(18,23)(19,26)(20,21)(24,32)(25,30)(27,31);(11,12)(14,16)(17,18)(19,27)(20,23)(21,30)(22,25)(24,26)(28,32)(29,31);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,28)(21,29)(22,26)(23,31)(25,27)(30,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(18,25)(19,26)(23,30)(28,29);(10,15)(14,16)(17,20,22,21)(18,23,25,30)(19,28,26,29)(24,31,27,32)

Polycyclic presentation:

2^9|c^2=hi,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6386834);