#9711955

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

4073 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=g,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,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, 9619151);