#511589

order	512
nilpotency class	4
isoclinism	isoclinic class of #511571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33171	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418785);