#493798

order	512
nilpotency class	4
isoclinism	isoclinic class of #481288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#418
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1804
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1011 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)(3,13,16,8)(4,18)(5,22)(6,28)(7,10)(9,20)(11,26)(12,25)(14,41,42,60)(15,45,23,56)(17,32,48,40)(19,51)(21,54)(24,38)(27,55)(29,37)(30,43,39,53)(31,49,36,58)(33,52)(34,59)(35,46)(44,62,50,64)(47,61,57,63);(1,3,6,17,7,16,25,48)(2,8,28,40,10,13,12,32)(4,14,29,50,19,42,21,44)(5,23,26,58,27,15,46,49)(9,30,34,61,33,39,38,63)(11,36,55,56,35,31,22,45)(18,41,37,64,51,60,54,62)(20,53,59,47,52,43,24,57);(1,4)(2,9)(3,14)(5,24)(6,29)(7,19)(8,30)(10,33)(11,37)(12,38)(13,39)(15,47)(16,42)(17,50)(18,22)(20,26)(21,25)(23,57)(27,59)(28,34)(31,62)(32,63)(35,54)(36,64)(40,61)(41,45)(43,49)(44,48)(46,52)(51,55)(53,58)(56,60);(2,10)(8,13)(9,33)(11,35)(12,28)(18,51)(22,55)(30,39)(31,36)(32,40)(34,38)(37,54)(41,60)(45,56)(61,63)(62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400994);