#190627

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
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	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

863 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=e,f^2=i,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97823);