#504084

order	512
nilpotency class	4
isoclinism	isoclinic class of #503866
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=ef,d^2=ei,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411280);