#206061

order	512
nilpotency class	3
isoclinism	isoclinic class of #205777
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3188	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=f,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=ch,e^b=ei,g^b=gi,d^c=di

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, 113257);