#9267063

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

4184 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)(3,8)(4,10)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,15,16,13)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(35,41)(38,45)(42,47)(46,48);(2,9)(3,4)(5,11)(13,15)(18,23)(19,26)(24,29)(27,32)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4,7,3)(2,10,9,8)(5,14,11,12)(6,15,16,13)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,13,10,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,14)(10,12)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,31)(26,32)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9174259);