#8726648

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

3801 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gi,e^2=i,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, 8633844);