#504423

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	#13331	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,d^2=fi,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,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, 411619);