#181275

order	512
nilpotency class	3
isoclinism	isoclinic class of #181134
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	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1398 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88471);