#223043

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

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1085 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ch,d^b=de,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, 130239);