#9290471

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

4009 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=gi,e^2=i,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, 9197667);