#494119

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,g^b=gi,g^c=gi

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, 401315);