#509252

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 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:

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

Polycyclic presentation:

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

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, 416448);