#494103

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#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×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

622 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)(4,6)(7,8)(9,10,14,21,15,19,32,39,16,22,33,40,13,20,24,34)(11,23,27,38,28,35,12,29,25,36,31,18,26,17,30,37);(1,3,4,7)(2,5,6,8)(9,11,16,25)(10,17)(12,24,30,32)(13,26,15,28)(14,27,33,31)(18,34)(19,23)(20,36)(21,37)(22,35)(29,40)(38,39);(2,6)(5,8)(9,12,13,27,16,30,15,31)(10,18,19,37,22,38,20,29)(11,24,26,33,25,32,28,14)(17,34,23,21,35,39,36,40);(11,25)(12,30)(17,35)(18,38)(23,36)(26,28)(27,31)(29,37)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=di,d^c=di,g^c=gi

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, 401299);