#9508938

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

3988 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=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, 9416134);