#501803

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	#24290	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

650 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,27)(13,16)(14,20)(15,21)(18,28)(19,32)(22,41)(23,52)(24,36)(25,40)(26,42)(29,56)(30,44)(31,45)(33,37)(34,48)(35,47)(38,50)(39,53)(43,57)(46,61)(49,62)(51,66)(54,64)(55,65)(58,67)(59,63)(60,69)(68,70)(71,72);(1,3,4,7)(2,5,6,8)(10,16)(11,22)(13,32)(14,33)(17,38)(18,27)(19,21)(20,47)(23,49)(24,34)(25,50)(26,51)(28,30)(29,58)(35,61)(36,41)(39,62)(40,66)(42,48)(43,67)(44,45)(52,63)(53,71)(54,70)(55,59)(57,60)(64,72)(65,68);(2,6)(5,8)(9,11)(10,17)(12,23)(13,24)(14,34)(15,26)(16,36)(18,39)(19,40)(20,48)(21,42)(22,33)(25,32)(27,52)(28,53)(29,59)(30,55)(31,54)(35,50)(37,41)(38,47)(43,68)(44,65)(45,64)(46,66)(49,58)(51,61)(56,63)(57,70)(60,71)(62,67)(69,72);(9,12,15,31)(10,18,21,44)(11,23,26,54)(13,30,32,28)(14,29,35,60)(16,27,19,45)(17,39,42,65)(20,43,47,67)(22,49,51,70)(24,55,25,53)(33,58,61,57)(34,59,50,71)(36,52,40,64)(37,56,46,69)(38,62,48,68)(41,63,66,72)

Polycyclic presentation:

2^9|b^2=f,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, 408999);