#504122

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 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:

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

Polycyclic presentation:

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

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, 411318);