#9730303

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

4185 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,25)(19,24,22,32)(20,26,29,21)(27,28,30,31)(33,34,40,39)(35,45,47,41)(36,42,37,43)(38,44,46,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,22)(18,24,25,32)(20,27,29,30)(21,31,26,28)(33,35,40,47)(34,41,39,45)(36,46,37,38)(42,44,43,48);(2,3)(7,9)(8,10)(14,15)(18,25)(20,29)(24,32)(27,30)(33,36,40,37)(34,42,39,43)(35,46,47,38)(41,44,45,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,26)(19,27)(21,25)(22,30)(23,29)(24,28)(31,32)(33,37,40,36)(34,43,39,42)(35,38,47,46)(41,48,45,44);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,26)(18,20,25,29)(19,28,22,31)(24,30,32,27)(33,38)(34,44)(35,36)(37,47)(39,48)(40,46)(41,43)(42,45);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,19)(18,24,25,32)(20,30,29,27)(21,31,26,28)(33,39,40,34)(35,45,47,41)(36,43,37,42)(38,44,46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=g,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9637499);