#8912165

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

4203 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)(3,7,10,4)(5,8,14,16)(11,13,15,12)(17,18,22,24)(19,27,29,23)(20,25,31,30)(21,28,32,26)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(1,3)(2,7)(4,9)(5,11)(6,10)(8,13)(12,16)(14,15)(17,19,22,29)(18,23,24,27)(20,26,31,28)(21,25,32,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(3,10)(4,7)(11,15)(12,13)(18,24)(21,32)(23,27)(25,30)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(1,4,6,7)(2,3,9,10)(5,13,14,12)(8,15,16,11)(17,18)(19,23)(20,30)(21,26)(22,24)(25,31)(27,29)(28,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,16)(10,15)(17,20,22,31)(18,25,24,30)(19,28,29,26)(21,27,32,23)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,3)(2,7)(4,9)(5,11)(6,10)(8,13)(12,16)(14,15)(17,21)(18,26)(19,25)(20,27)(22,32)(23,31)(24,28)(29,30)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8819361);