#9020323

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	#48647	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=gh,e^2=g,f^2=gi,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, 8927519);