#149119

order	512
nilpotency class	4
isoclinism	isoclinic class of #13545
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2101
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

421 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56315);