#8873616

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

4096 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:

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

Polycyclic presentation:

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

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, 8780812);