#6447058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6447056
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2420
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354254);