#504120

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

512 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:

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

Polycyclic presentation:

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

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, 411316);