#494434

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#386
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1869
prime radical	#3449	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1277 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,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, 401630);