#493655

order	512
nilpotency class	4
isoclinism	isoclinic class of #493653
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20609
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#63063	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2158 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)(3,15)(4,20,6,12)(5,11,25,37)(7,13)(8,14)(9,16,32,26)(10,22,34,27)(17,41,45,58)(18,29)(19,38)(21,48)(23,35)(24,36,52,60)(28,40)(30,50)(31,43,39,46)(33,51)(42,56)(44,53,63,64)(47,54)(49,57)(55,62)(59,61);(1,3,8,19)(2,9,14,32)(4,16,6,26)(5,17,25,45)(7,18,28,47)(10,29,34,54)(11,30,37,56)(12,38,20,15)(13,39,40,31)(21,42,23,50)(22,43,27,46)(24,44,52,63)(33,53,57,64)(35,58,48,41)(36,59,60,55)(49,61,51,62);(1,4)(2,10)(3,16)(5,21)(6,8)(7,27)(9,29)(11,33)(12,13)(14,34)(15,31)(17,42)(18,46)(19,26)(20,40)(22,28)(23,25)(24,51)(30,53)(32,54)(35,36)(37,57)(38,39)(41,55)(43,47)(44,62)(45,50)(48,60)(49,52)(56,64)(58,59)(61,63);(1,5)(2,11)(3,17)(4,21)(6,23)(7,24)(8,25)(9,30)(10,33)(12,35)(13,36)(14,37)(15,41)(16,42)(18,44)(19,45)(20,48)(22,49)(26,50)(27,51)(28,52)(29,53)(31,55)(32,56)(34,57)(38,58)(39,59)(40,60)(43,61)(46,62)(47,63)(54,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400851);