#493347

order	512
nilpotency class	3
isoclinism	isoclinic class of #1795
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#379
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1779
prime radical	C₂×#585	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3137 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,e^b=ei,g^b=gi,d^c=di,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, 400543);