#500462

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61289	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

761 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=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, 407658);