#509014

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
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

531 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=i,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,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, 416210);