#181294

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

1528 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)(9,10,13,19,15,18,14,20)(11,21,23,34,24,16,28,30)(12,17,26,32,25,31,27,33)(22,35,37,40,36,29,38,39);(2,5,7,6)(9,11,14,23,15,24,13,28)(10,16,20,30,18,21,19,34)(12,22,27,37,25,36,26,38)(17,29,33,39,31,35,32,40);(9,12)(10,17)(11,22)(13,26)(14,27)(15,25)(16,29)(18,31)(19,32)(20,33)(21,35)(23,37)(24,36)(28,38)(30,39)(34,40);(12,25)(17,31)(22,36)(26,27)(29,35)(32,33)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,f^2=i,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, 88490);