#501807

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

Polycyclic presentation:

2^9|b^2=fi,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, 409003);