#9783009

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

4317 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,e^2=g,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, 9690205);