#9545828

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

4160 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)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(18,19)(23,25)(24,26)(30,31)(33,34,39,42)(35,40,37,38)(36,41,45,48)(43,47,46,44);(2,6)(7,9)(8,12)(14,15)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35,39,37)(34,40,42,38)(36,43,45,46)(41,47,48,44);(2,6)(3,10)(4,13)(14,15)(20,27)(21,29)(23,25)(24,26)(35,37)(38,40)(43,46)(44,47);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,36,39,45)(34,41,42,48)(35,43,37,46)(38,44,40,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,37,39,35)(34,38,42,40)(36,43,45,46)(41,47,48,44);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,35)(36,44)(37,42)(39,40)(41,43)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=gi,e^2=i,f^2=h,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, 9453024);