#498341

order	512
nilpotency class	4
isoclinism	isoclinic class of #498338
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23001
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

596 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=ei,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fhi,g^c=gi,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, 405537);