#509154

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=i,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416350);