#9730643

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
prime radical	#55449	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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)(3,12)(4,9)(5,8)(6,10)(7,11)(13,14)(15,16)(17,18)(19,21)(20,29)(22,31)(23,24)(25,30)(26,28)(27,32)(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)(17,19,23,27)(18,21,24,32)(20,28,30,22)(25,31,29,26)(35,41)(38,45)(42,47)(46,48);(18,24)(19,27)(20,30)(26,31)(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)(18,25)(19,28)(21,31)(22,27)(23,30)(24,29)(26,32)(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)(20,31)(22,29)(23,32)(24,27)(25,28)(26,30)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,6,7,9)(2,10,11,4)(3,14,5,16)(8,15,12,13)(17,22,23,28)(18,26,24,31)(19,20,27,30)(21,25,32,29)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=g,e^2=g,f^2=gh,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, 9637839);