#6480608

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1556 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)(11,12)(14,16)(17,18)(19,20)(21,27)(22,28)(23,32)(24,29)(25,30)(26,31);(9,10)(11,16)(12,14)(13,15)(17,19,23,30,24,26,22,27)(18,25,28,20,29,21,32,31);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20)(18,26)(19,29)(21,23)(22,25)(24,31)(27,28)(30,32);(10,15)(14,16)(17,21)(18,27)(19,28)(20,22)(23,31)(24,25)(26,32)(29,30);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,22,24,23)(18,28,29,32)(19,27,26,30)(20,21,31,25)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6387804);