#8622339

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

4197 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=h,e^2=i,f^2=gh,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, 8529535);