#223082

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=ch,d^b=dei,f^c=fi,g^c=ghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130278);