#496181

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

997 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:

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

Polycyclic presentation:

2^9|a^2=g,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,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, 403377);