#8873597

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

4020 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18)(19,20)(21,29)(22,26)(23,24)(25,30)(27,31)(28,32)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(2,7)(5,9)(8,10)(13,16)(17,18,22,26)(19,23,24,20)(21,29,30,25)(27,28,32,31)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(3,11)(8,10)(12,14)(13,16)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,3,6,11)(2,8,7,10)(4,14,15,12)(5,13,9,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,5,6,9)(2,4,7,15)(3,13,11,16)(8,14,10,12)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=ghi,e^2=hi,f^2=gh,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, 8780793);