#223070

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

1248 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)(20,24)(21,23)(22,27)(25,29)(28,30)(31,32);(1,2)(3,6)(4,5)(7,8)(11,16)(13,15)(17,19)(18,22)(20,27)(21,28)(23,31)(24,26)(25,30)(29,32);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,13,14,15)(18,23)(19,27)(21,29)(22,30)(24,25)(26,31);(2,5)(6,8)(10,14)(13,15)

Polycyclic presentation:

2^9|a^2=g,c^2=eh,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, 130266);