#501809

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	#24291	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

667 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)(3,5)(4,6)(7,8)(9,10)(11,22)(12,27)(13,16)(14,21)(15,20)(17,24)(18,30)(19,32)(23,35)(25,38)(26,28)(29,33)(31,36)(34,37)(39,40);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,28)(13,23)(14,31)(15,25)(16,34)(18,35)(19,38)(20,36)(21,22)(24,32)(26,33)(27,40)(29,37)(30,39);(2,6)(5,8)(9,12,15,13)(10,16,20,27)(11,22,25,38)(14,29,32,30)(17,24,36,31)(18,19,33,21)(23,34,28,40)(26,37,35,39);(9,13,15,12)(10,18,20,33)(11,23,25,28)(14,29,32,30)(16,19,27,21)(17,35,36,26)(22,34,38,40)(24,39,31,37)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,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, 409005);