#497412

order	512
nilpotency class	4
isoclinism	isoclinic class of #497408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3293	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404608);