#9730305

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

3939 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,24,23)(19,28,22,25)(20,26,21,27)(29,32,30,31)(33,34)(35,44)(36,42)(37,43)(38,41)(39,40)(45,47)(46,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,24,22)(18,25,23,28)(20,29,21,30)(26,31,27,32)(34,40)(37,47)(41,44)(42,48);(2,3)(7,9)(8,10)(14,15)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,35,39,38)(34,41,40,44)(36,43,46,45)(37,42,47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,36,39,46)(34,42,40,48)(35,45,38,43)(37,44,47,41);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,22,24,19)(18,25,23,28)(20,30,21,29)(26,31,27,32)(33,37)(34,43)(35,42)(36,44)(38,48)(39,47)(40,45)(41,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,38,39,35)(34,44,40,41)(36,43,46,45)(37,42,47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=g,e^2=ghi,f^2=g,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, 9637501);