#511693

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

487 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,d^2=ei,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, 418889);