#509466

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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)(12,29)(13,19)(14,17)(15,21)(16,25)(18,34)(20,35)(23,51)(24,55)(26,40)(27,50)(28,41)(30,60)(31,42)(32,45)(33,59)(36,47)(37,43)(38,54)(39,56)(44,58)(46,65)(48,66)(49,53)(52,67)(57,71)(61,69)(62,72)(63,68)(64,70);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,30)(13,25)(14,26)(15,23)(17,41)(18,44)(19,22)(20,40)(21,49)(24,31)(27,37)(28,35)(29,55)(32,58)(33,61)(34,56)(36,53)(38,43)(39,45)(42,60)(46,71)(47,51)(48,50)(52,63)(54,66)(57,64)(59,67)(62,65)(68,69)(70,72);(1,4,5,8)(2,6,3,7)(10,17)(11,23)(14,35)(15,37)(16,38)(18,42)(19,20)(21,47)(22,50)(24,52)(25,53)(26,54)(27,28)(29,45)(30,61)(32,34)(33,65)(36,66)(39,67)(40,51)(41,49)(44,69)(46,70)(55,72)(56,57)(58,62)(60,71)(63,64);(2,3)(6,7)(9,12)(10,18)(11,24)(13,31)(14,34)(15,33)(16,39)(17,42)(19,45)(20,29)(21,46)(22,44)(23,52)(25,30)(26,58)(27,57)(28,56)(32,35)(36,63)(37,65)(38,67)(40,60)(41,55)(43,68)(47,70)(48,59)(49,72)(50,69)(51,71)(53,61)(54,62)(64,66)

Polycyclic presentation:

2^9|b^2=ei,c^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416662);