#504117

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

529 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)(11,21)(12,14)(13,28)(15,19)(16,20)(17,23)(18,27)(22,24)(25,35)(26,36)(29,33)(30,31)(32,37)(34,39)(38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,22)(13,25)(14,32)(15,23)(16,26)(18,24)(19,36)(20,35)(21,28)(27,34)(29,38)(30,37)(31,40)(33,39);(1,4,5,3)(2,6,7,8)(9,12)(10,14)(11,21)(13,27)(15,30)(16,29)(17,25)(18,28)(19,31)(20,33)(22,32)(23,35)(24,37)(26,36)(34,40)(38,39);(9,12)(10,18)(11,22)(13,30)(14,28)(15,27)(16,29)(17,24)(19,33)(20,31)(21,32)(23,34)(25,37)(26,38)(35,40)(36,39)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,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, 411313);