#224149

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

895 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)(6,7)(10,14,15,16)(17,18)(19,22)(20,27,29,23)(21,24)(25,32,28,30)(26,31);(1,2,4,5)(3,7,8,6)(17,19,21,26)(18,22,24,31)(20,28,29,25)(23,32,27,30);(9,10)(11,14)(12,16)(13,15)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,14,15,16)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=ei,f^2=i,b^a=be,c^a=cf,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, 131345);