#8691762

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

4224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8598958);