#493649

order	512
nilpotency class	4
isoclinism	isoclinic class of #493638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20609
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#63063	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2161 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,f^b=fhi,g^b=gh,e^c=eh,g^d=gi,f^e=fi,g^e=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400845);