#493954

order	512
nilpotency class	3
isoclinism	isoclinic class of #392203
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2280
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#20715
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1033 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401150);