#504471

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
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×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

569 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)(3,6)(4,5)(7,8)(9,10,13,20,15,22,38,48,14,21,37,54,16,23,39,55)(11,24,27,44,29,60,68,43,28,59,67,45,30,52,35,17)(12,31,32,56,40,50,69,51,33,18,46,49,34,53,36,19)(25,61,62,47,65,71,72,41,63,57,66,58,26,42,64,70);(9,11,14,28)(10,17,21,43)(12,26,33,65)(13,35,37,68)(15,30,16,29)(18,47,31,58)(19,42,51,71)(20,52,54,60)(22,45,23,44)(24,48,59,55)(25,40,63,34)(27,38,67,39)(32,66,46,62)(36,64,69,72)(41,53,70,50)(49,61,56,57);(2,5)(6,8)(9,12)(10,18)(11,25)(13,32)(14,33)(15,40)(16,34)(17,41)(19,48)(20,49)(21,31)(22,53)(23,50)(24,57)(26,30)(27,62)(28,63)(29,65)(35,64)(36,39)(37,46)(38,69)(42,60)(43,70)(44,58)(45,47)(51,55)(52,71)(54,56)(59,61)(66,67)(68,72);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,26)(13,36)(14,33)(15,34)(16,40)(17,42)(18,48)(20,53)(21,51)(22,49)(23,56)(24,58)(25,30)(27,66)(28,65)(29,63)(31,55)(32,39)(35,64)(37,69)(38,46)(41,60)(43,71)(44,57)(45,61)(47,59)(50,54)(52,70)(62,67)(68,72)

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=fi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi

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, 411667);