#509936

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
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

513 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)(9,10,16,18,15,22,13,20)(11,17,27,41,28,42,25,39)(12,29,32,19,33,45,30,46)(14,21,36,43,35,49,34,47)(23,48,51,37,52,63,26,40)(24,53,57,38,56,66,54,65)(31,58,60,44,61,68,59,69)(50,62,55,67,71,64,70,72);(1,3,5,2)(4,7,8,6)(9,11)(10,17)(12,24)(13,25)(14,23)(15,28)(16,27)(18,41)(19,38)(20,39)(21,48)(22,42)(26,34)(29,53)(30,54)(31,50)(32,57)(33,56)(35,52)(36,51)(37,43)(40,47)(44,67)(45,66)(46,65)(49,63)(55,60)(58,62)(59,70)(61,71)(64,68)(69,72);(1,4,5,8)(2,6,3,7)(10,18)(11,23)(13,16)(14,34)(17,37)(19,29)(20,22)(24,50)(25,51)(26,27)(28,52)(30,32)(31,60)(35,36)(38,62)(39,63)(40,42)(41,48)(43,47)(45,46)(53,67)(54,55)(56,71)(57,70)(58,68)(59,61)(64,65)(66,72);(2,3)(6,7)(9,12,15,33)(10,19,22,46)(11,24,28,56)(13,32,16,30)(14,31,35,61)(17,38,42,65)(18,29,20,45)(21,44,49,69)(23,50,52,71)(25,57,27,54)(26,55,51,70)(34,60,36,59)(37,62,40,64)(39,66,41,53)(43,58,47,68)(48,67,63,72)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=e,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417132);