#8726403

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,5)(3,8,10,4)(6,9,15,13)(11,14,16,12)(17,18,23,24)(19,22,25,32)(20,21,29,30)(26,31,28,27)(33,34)(35,44)(36,46)(37,45)(38,40)(39,43)(41,47)(42,48);(1,3,7,10)(2,8,5,4)(6,11,15,16)(9,14,13,12)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31)(33,35,39,38)(34,40,43,44)(36,42,47,45)(37,41,48,46);(3,10)(4,8)(11,16)(12,14)(18,24)(19,25)(20,29)(27,31)(33,35,39,38)(34,40,43,44)(36,42,47,45)(37,41,48,46);(1,4)(2,3)(5,10)(6,14)(7,8)(9,16)(11,13)(12,15)(17,20)(18,21)(19,26)(22,31)(23,29)(24,30)(25,28)(27,32)(33,36,39,47)(34,41,43,46)(35,45,38,42)(37,44,48,40);(1,5,7,2)(3,4,10,8)(6,9,15,13)(11,14,16,12)(17,21)(18,20)(19,27)(22,28)(23,30)(24,29)(25,31)(26,32)(33,37)(34,42)(35,41)(36,44)(38,46)(39,48)(40,47)(43,45);(1,6,7,15)(2,9,5,13)(3,11,10,16)(4,12,8,14)(17,22,23,32)(18,25,24,19)(20,27,29,31)(21,26,30,28)(33,38,39,35)(34,40,43,44)(36,45,47,42)(37,41,48,46)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633599);