#498974

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

914 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)(11,17)(12,27)(13,19)(14,34)(15,38)(16,21)(18,30)(20,36)(22,50)(23,40)(24,55)(25,42)(26,43)(28,59)(29,48)(31,64)(32,61)(33,49)(35,47)(37,60)(39,52)(41,57)(44,56)(45,62)(46,63)(51,68)(53,71)(54,70)(58,69)(65,67)(66,72);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,28)(13,23)(14,24)(15,25)(16,26)(18,44)(19,40)(20,41)(21,43)(22,29)(27,59)(30,56)(31,53)(32,51)(33,54)(34,55)(35,65)(36,57)(37,58)(38,42)(39,45)(46,66)(47,67)(48,50)(49,70)(52,62)(60,69)(61,68)(63,72)(64,71);(9,12,16,32)(10,18,21,46)(11,22,26,53)(13,31,15,29)(14,35,37,62)(17,39,43,67)(19,47,38,45)(20,48,49,64)(23,28,25,51)(24,56,58,72)(27,60,61,34)(30,33,63,36)(40,44,42,66)(41,68,70,59)(50,69,71,55)(52,54,65,57);(9,13,16,15)(10,19,21,38)(11,23,26,25)(12,29,32,31)(14,33,37,36)(17,40,43,42)(18,45,46,47)(20,34,49,60)(22,51,53,28)(24,54,58,57)(27,48,61,64)(30,62,63,35)(39,66,67,44)(41,55,70,69)(50,68,71,59)(52,72,65,56)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406170);