#497454

order	512
nilpotency class	4
isoclinism	isoclinic class of #497451
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²⋊C₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3415	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

958 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:

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

Polycyclic presentation:

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

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, 404650);