#9783308

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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)(17,18,23,27)(19,24,29,21)(20,30,28,25)(22,32,31,26)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,47,44);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,24)(20,31)(21,27)(22,28)(23,29)(25,32)(26,30)(35,43)(36,45)(39,46)(40,48);(2,3)(7,9)(8,10)(14,15)(17,20,23,28)(18,25,27,30)(19,22,29,31)(21,32,24,26)(33,35)(34,39)(36,44)(37,40)(38,43)(41,45)(42,46)(47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,21,23,24)(18,19,27,29)(20,26,28,32)(22,25,31,30);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,19)(18,24)(20,22)(21,27)(23,29)(25,26)(28,31)(30,32)(33,36)(34,40)(35,44)(37,39)(38,45)(41,43)(42,48)(46,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22)(18,26)(19,28)(20,29)(21,25)(23,31)(24,30)(27,32)(33,37)(34,41)(35,40)(36,46)(38,47)(39,45)(42,44)(43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=h,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, 9690504);