#497925

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	C₂²×#107
prime radical	C₂²×D₄	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

573 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:

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

Polycyclic presentation:

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

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, 405121);