#504123

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×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 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)(9,10,15,21)(11,23,26,36)(12,16,13,18)(14,22,32,20)(17,31,34,25)(19,29,38,30)(24,27,28,40)(33,35,37,39);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,28)(13,24)(14,31)(15,26)(16,33)(18,37)(19,27)(20,36)(21,34)(22,23)(25,32)(29,35)(30,39)(38,40);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,23)(13,15)(14,29)(16,21)(17,31)(19,22)(20,38)(24,37)(25,34)(26,36)(27,35)(28,33)(30,32)(39,40);(9,13)(10,19)(11,24)(12,15)(14,29)(16,20)(17,27)(18,22)(21,38)(23,37)(25,39)(26,28)(30,32)(31,35)(33,36)(34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fi,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, 411319);