#505442

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

446 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,20,22)(19,23,26,30)(21,24,28,29)(25,31,32,27);(2,6)(7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,26)(21,25)(22,30)(24,31)(27,29)(28,32);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,15,16,14)(18,22)(19,25)(21,29)(23,27)(24,28)(26,32)(30,31);(1,4,5,3)(2,7,6,8)(10,16)(14,15)(17,18,20,22)(19,23,26,30)(21,24,28,29)(25,31,32,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=e,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,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, 412638);