#9685621

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

4239 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9592817);