#498619

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

520 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405815);