#8479527

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

4196 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18)(19,23)(20,29)(21,24)(22,25)(26,32)(27,30)(28,31)(33,34)(35,43)(36,37)(38,40)(39,42)(41,47)(44,46)(45,48);(2,4)(7,9)(8,12)(15,16)(17,19,22,28)(18,23,25,31)(20,27,21,26)(24,32,29,30)(33,35,39,38)(34,40,42,43)(36,44,47,45)(37,48,41,46);(17,20,22,21)(18,24,25,29)(19,26,28,27)(23,30,31,32)(33,36)(34,41)(35,44)(37,42)(38,45)(39,47)(40,46)(43,48);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,21,22,20)(18,24,25,29)(19,27,28,26)(23,30,31,32)(33,36)(34,41)(35,45)(37,42)(38,44)(39,47)(40,48)(43,46);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,39,41)(34,36,42,47)(35,46,38,48)(40,45,43,44);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,32)(28,31)(33,38,39,35)(34,40,42,43)(36,45,47,44)(37,48,41,46)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gh,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, 8386723);