#496183

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2308
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21738
prime radical	#3449	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

993 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403379);