#9206510

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

3939 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)(3,10)(4,12)(5,8)(6,7)(9,14)(11,15)(13,16)(17,18,22,25)(19,26,21,23)(20,24,29,32)(27,31,28,30)(34,36)(35,42)(38,46)(45,47);(3,5)(8,10)(11,13)(15,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,34,39,36)(35,37,42,40)(38,41,46,44)(43,45,48,47);(1,3,7,5)(2,8,6,10)(4,13,14,11)(9,16,12,15)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(33,35,39,42)(34,40,36,37)(38,43,46,48)(41,47,44,45);(1,4,7,14)(2,9,6,12)(3,11,5,13)(8,15,10,16)(17,20,22,29)(18,24,25,32)(19,27,21,28)(23,30,26,31)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,5,7,3)(2,8,6,10)(4,11,14,13)(9,16,12,15)(17,21,22,19)(18,23,25,26)(20,27,29,28)(24,31,32,30)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,6,7,2)(3,10,5,8)(4,12,14,9)(11,16,13,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9113706);