#496026

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

534 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)(9,10,14,20,15,19,25,36,16,21,24,17,13,31,11,22)(12,18,28,33,29,34,26,39,30,35,38,32,27,40,23,37);(2,5)(6,8)(9,11,13,24,16,25,15,14)(10,17,19,22,21,20,31,36)(12,26,27,28,30,23,29,38)(18,33,34,39,35,32,40,37);(1,3,4,7)(2,6,5,8)(9,11,13,24,16,25,15,14)(10,17,19,22,21,20,31,36)(12,26,27,28,30,23,29,38)(18,33,34,39,35,32,40,37);(9,12,14,28,15,29,25,26,16,30,24,38,13,27,11,23)(10,18,20,33,19,34,36,39,21,35,17,32,31,40,22,37)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=g,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=di,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, 403222);