#9783315

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3889 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9690511);