#181226

order	512
nilpotency class	3
isoclinism	isoclinic class of #181129
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1574 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)(4,7)(6,8)(9,10)(11,21,24,16)(12,17)(13,18)(14,20)(15,19)(22,35,37,29)(23,31,25,30)(26,32)(27,33)(28,34)(36,40,38,39);(2,5,7,6)(9,11,14,25,15,24,13,23)(10,16,20,31,19,21,18,30)(12,22,26,36,27,37,28,38)(17,29,32,39,33,35,34,40);(9,12,15,27)(10,17,19,33)(11,22,24,37)(13,28,14,26)(16,29,21,35)(18,34,20,32)(23,38,25,36)(30,40,31,39);(9,13,15,14)(10,18,19,20)(11,23,24,25)(12,26,27,28)(16,30,21,31)(17,32,33,34)(22,36,37,38)(29,39,35,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=be,e^a=eh,g^a=ghi,f^b=fi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88422);