#493848

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#381
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1817
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1105 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:

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

Polycyclic presentation:

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

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, 401044);