#9545922

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

4000 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(33,34,39,42)(35,40,37,38)(36,41,45,48)(43,47,46,44);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32)(33,35,39,37)(34,40,42,38)(36,43,45,46)(41,47,48,44);(2,7)(3,10)(8,15)(11,12)(18,22)(19,21)(20,28)(31,32)(35,37)(38,40)(43,46)(44,47);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,23,28)(18,25,22,29)(19,27,21,30)(24,31,26,32)(33,36,39,45)(34,41,42,48)(35,43,37,46)(38,44,40,47);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,21,23,19)(18,26,22,24)(20,27,28,30)(25,31,29,32)(33,37,39,35)(34,38,42,40)(36,43,45,46)(41,47,48,44);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,22,23,18)(19,26,21,24)(20,29,28,25)(27,32,30,31)(33,38)(34,35)(36,44)(37,42)(39,40)(41,43)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=gi,e^2=i,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, 9453118);