#224593

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

934 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=e,d^2=eh,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, 131789);