#9730657

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

4092 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)(3,12)(4,9)(5,8)(6,10)(7,11)(13,14)(15,16)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3,7,5)(2,8,11,12)(4,13,10,15)(6,14,9,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(35,41)(38,45)(42,47)(46,48);(18,20)(19,26)(25,28)(27,32)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4,7,10)(2,9,11,6)(3,13,5,15)(8,16,12,14)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5,7,3)(2,8,11,12)(4,15,10,13)(6,14,9,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,6,7,9)(2,10,11,4)(3,14,5,16)(8,15,12,13)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=g,e^2=gh,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, 9637853);