#8648960

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, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8556156);