#9267016

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

3992 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=gi,e^2=h,f^2=g,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, 9174212);