#493154

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,e^2=hi,f^2=hi,h^2=i,b^a=be,d^a=di,e^b=eh,f^b=fh,h^b=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400350);