#8429381

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

4179 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8336577);