#396362

order	512
nilpotency class	3
isoclinism	isoclinic class of #396361
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14732
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1381 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)(3,15)(4,10)(5,26,28,39)(6,13)(7,12)(8,14)(9,18)(11,29,38,27)(16,41)(17,45,47,31)(19,32)(20,40)(21,30)(22,51,53,55)(23,36)(24,35)(25,37)(33,44)(34,52,59,54)(42,48,60,46)(43,56,62,63)(49,57)(50,58,64,61);(1,3,6,18,8,20,7,19)(2,9,12,15,14,32,13,40)(4,21,23,44,25,49,24,16)(5,17,27,46,28,47,29,48)(10,33,35,30,37,41,36,57)(11,31,26,42,38,45,39,60)(22,50,52,62,53,64,54,43)(34,58,55,56,59,61,51,63);(1,4,8,25)(2,10,14,37)(3,16,20,44)(5,22,28,53)(6,23,7,24)(9,30,32,57)(11,34,38,59)(12,35,13,36)(15,41,40,33)(17,43,47,62)(18,21,19,49)(26,55,39,51)(27,52,29,54)(31,56,45,63)(42,61,60,58)(46,50,48,64);(1,5,6,27,8,28,7,29)(2,11,12,26,14,38,13,39)(3,17,18,46,20,47,19,48)(4,22,23,52,25,53,24,54)(9,31,15,42,32,45,40,60)(10,34,35,55,37,59,36,51)(16,43,21,50,44,62,49,64)(30,56,41,61,57,63,33,58)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303558);