#8718318

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

4132 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625514);