#493687

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1797
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1161 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400883);