#220185

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×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1091 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^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, 127381);