#9127598

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

4121 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,23,22)(19,26,21,24)(20,28,29,25)(27,31,30,32)(33,34)(35,38)(36,41)(37,42)(39,40)(43,48)(44,46)(45,47);(2,7)(4,15)(8,11)(10,13)(17,19,23,21)(18,24,22,26)(20,27,29,30)(25,31,28,32)(33,35,39,43)(34,38,40,48)(36,44,45,37)(41,46,47,42);(2,7)(3,12)(9,14)(10,13)(20,29)(25,28)(27,30)(31,32)(34,40)(35,43)(36,45)(42,46);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,23,29)(18,25,22,28)(19,27,21,30)(24,31,26,32)(33,36)(34,41)(35,44)(37,43)(38,46)(39,45)(40,47)(42,48);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,21,23,19)(18,24,22,26)(20,27,29,30)(25,32,28,31)(33,37,39,44)(34,42,40,46)(35,45,43,36)(38,47,48,41);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,18)(19,26,21,24)(20,28,29,25)(27,32,30,31)(33,38)(34,43)(35,40)(36,46)(37,41)(39,48)(42,45)(44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,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=dgi,e^d=egi,f^e=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, 9034794);