#504474

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

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,17,40,14,22,38,50,16,24,13,21,15,23,37,57)(11,25,59,47,29,61,35,18,31,62,28,46,30,56,41,48)(12,19,42,54,39,58,36,20,34,53,32,51,33,52,43,55)(26,60,67,49,66,71,65,44,64,70,63,72,27,45,68,69);(9,11,16,31)(10,18,24,48)(12,27,34,66)(13,35,17,41)(14,30,15,29)(19,49,53,72)(20,45,55,71)(21,56,40,61)(22,47,23,46)(25,57,62,50)(26,39,64,33)(28,37,59,38)(32,67,42,63)(36,65,43,68)(44,58,69,52)(51,70,54,60);(2,5)(6,8)(9,12)(10,19)(11,26)(13,32)(14,39)(15,33)(16,34)(17,42)(18,44)(20,50)(21,51)(22,58)(23,52)(24,53)(25,60)(27,30)(28,63)(29,66)(31,64)(35,65)(36,38)(37,43)(40,54)(41,68)(45,56)(46,72)(47,49)(48,69)(55,57)(59,67)(61,71)(62,70);(1,3,4,7)(2,6,5,8)(9,12)(10,20)(11,27)(13,36)(14,33)(15,39)(16,34)(17,43)(18,45)(19,50)(21,52)(22,54)(23,51)(24,55)(25,49)(26,30)(28,67)(29,64)(31,66)(32,38)(35,65)(37,42)(40,58)(41,68)(44,56)(46,70)(47,60)(48,71)(53,57)(59,63)(61,69)(62,72)

Polycyclic presentation:

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

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, 411670);