#9726577

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,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, 9633773);