#9026899

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	#56295	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 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:

(1,2,6,9)(3,7,11,5)(4,8,14,16)(10,15,12,13)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,34,40,39)(35,43,37,41)(36,42,46,48)(38,44,45,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,14)(13,16)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,35,40,37)(34,41,39,43)(36,45,46,38)(42,44,48,47);(1,3,6,11)(2,7,9,5)(4,12,14,10)(8,13,16,15)(17,18,22,26)(19,24,20,23)(21,29,30,25)(27,31,28,32)(33,36,40,46)(34,42,39,48)(35,45,37,38)(41,44,43,47);(1,3)(2,7)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,20,22,19)(18,24,26,23)(21,28,30,27)(25,32,29,31)(33,37,40,35)(34,43,39,41)(36,45,46,38)(42,44,48,47);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,31)(29,30)(33,38,40,45)(34,44,39,47)(35,36,37,46)(41,48,43,42);(1,5,6,7)(2,3,9,11)(4,13,14,15)(8,10,16,12)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)(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=gh,c^2=gi,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, 8934095);