#509424

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

488 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)(3,5,7,8)(9,10)(11,16)(12,28)(13,17)(14,20)(15,22)(18,29)(19,33)(21,35)(23,47)(24,51)(25,41)(26,39)(27,40)(30,46)(31,61)(32,58)(34,48)(36,43)(37,50)(38,52)(42,62)(44,59)(45,60)(49,63)(53,70)(54,71)(55,68)(56,65)(57,64)(66,69)(67,72);(1,2)(3,5)(4,6)(7,8)(9,11,15,27)(10,16,22,40)(12,24,32,54)(13,25,36,55)(14,23,35,26)(17,41,43,68)(18,38,45,67)(19,37,42,64)(20,47,21,39)(28,51,58,71)(29,52,60,72)(30,49,31,53)(33,50,62,57)(34,56,59,69)(44,66,48,65)(46,63,61,70);(1,3,4,7)(2,5,6,8)(10,17)(11,23)(13,33)(14,35)(16,37)(18,28)(19,21)(20,42)(22,43)(24,49)(25,55)(26,27)(29,34)(30,31)(36,62)(38,63)(39,68)(40,64)(41,47)(44,61)(45,58)(46,48)(51,66)(53,54)(56,69)(59,60)(65,71)(67,70);(2,6)(5,8)(9,12)(10,18)(11,24)(13,34)(14,30)(15,32)(16,38)(17,28)(19,46)(20,44)(21,48)(22,45)(23,49)(25,56)(26,53)(27,54)(29,33)(31,35)(36,59)(37,63)(39,65)(40,67)(41,51)(42,61)(43,58)(47,66)(50,52)(55,69)(57,72)(60,62)(64,70)(68,71)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=e,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, 416620);