#494775

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
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	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ 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) = 64

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

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

Polycyclic presentation:

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

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, 401971);