#9206396

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

4077 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,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, 9113592);