#220188

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,7,5,8,3,6)(9,10)(11,19)(12,24,26,16)(13,27)(14,18)(15,22)(17,28)(20,34,37,33)(21,31)(23,30)(25,32,40,39)(29,36,38,35);(1,3,5,4)(2,6,8,7)(9,11,14,23)(10,15,18,31)(12,20,26,37)(13,21,28,22)(16,29,24,38)(17,30,27,19)(25,35,40,36)(32,34,39,33);(9,12,14,26)(10,16,18,24)(11,20,23,37)(13,25,28,40)(15,29,31,38)(17,32,27,39)(19,33,30,34)(21,35,22,36);(11,21)(13,28)(15,30)(17,27)(19,31)(20,35)(22,23)(25,40)(29,34)(32,39)(33,38)(36,37)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,g^c=gh,e^d=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127384);