#8479524

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

4207 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)(3,7)(4,13)(5,8)(6,9)(10,16)(11,14)(12,15)(17,18,24,22)(19,28,23,25)(20,27,21,26)(29,31,30,32)(33,34,39,43)(35,44,48,40)(36,37,47,41)(38,42,45,46);(1,3,6,12)(2,7,9,15)(4,11,5,10)(8,16,13,14)(17,19,24,23)(18,25,22,28)(20,29,21,30)(26,31,27,32)(33,35)(34,40)(36,45)(37,42)(38,47)(39,48)(41,46)(43,44);(1,4,6,5)(2,8,9,13)(3,10,12,11)(7,14,15,16)(17,20,24,21)(18,26,22,27)(19,29,23,30)(25,31,28,32)(33,34,39,43)(35,40,48,44)(36,41,47,37)(38,42,45,46);(1,5,6,4)(2,8,9,13)(3,11,12,10)(7,14,15,16)(17,21,24,20)(18,27,22,26)(19,29,23,30)(25,31,28,32)(33,36)(34,41)(35,45)(37,43)(38,48)(39,47)(40,46)(42,44);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,22,24,18)(19,25,23,28)(20,27,21,26)(29,31,30,32)(33,37,39,41)(34,36,43,47)(35,46,48,42)(38,44,45,40);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,16)(12,15)(17,23,24,19)(18,25,22,28)(20,30,21,29)(26,31,27,32)(33,38,39,45)(34,42,43,46)(35,47,48,36)(37,44,41,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386720);