#509980

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
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×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

494 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)(9,10,16,36,35,53,33,18,15,22,13,20,14,21,38,48)(11,17,37,40,26,43,55,46,27,45,25,42,23,44,61,66)(12,28,62,49,30,52,34,51,32,47,29,50,31,63,39,19)(24,56,72,71,54,70,60,69,59,67,57,68,58,65,64,41);(1,3,5,2)(4,7,8,6)(9,11)(10,17)(12,24)(13,25)(14,23)(15,27)(16,37)(18,46)(19,41)(20,42)(21,44)(22,45)(26,35)(28,56)(29,57)(30,54)(31,58)(32,59)(33,55)(34,60)(36,40)(38,61)(39,64)(43,53)(47,67)(48,66)(49,71)(50,68)(51,69)(52,70)(62,72)(63,65);(1,4,5,8)(2,6,3,7)(10,18)(11,23)(13,33)(14,35)(16,38)(17,40)(19,47)(20,21)(22,48)(24,54)(25,37)(26,27)(28,51)(29,34)(30,31)(36,53)(39,62)(41,65)(42,45)(43,66)(44,46)(49,52)(50,63)(56,68)(58,59)(60,64)(67,71)(69,70);(2,3)(6,7)(9,12,15,32)(10,19,22,51)(11,24,27,59)(13,34,16,39)(14,30,35,31)(17,41,45,69)(18,47,48,28)(20,52,36,63)(21,49,53,50)(23,54,26,58)(25,60,37,64)(29,38,62,33)(40,65,42,70)(43,68,44,71)(46,67,66,56)(55,57,61,72)

Polycyclic presentation:

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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417176);