#9015394

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

4084 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922590);