#493343

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#120
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1779
prime radical	C₂×#585	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3143 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341701
maximal subgroup of	#72281241, #72281308, #72281337, …
maximal normal subgroup of	#72281241, #72281308, #72281337, …
maximal quotient of	#72287551

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400539);