#8827947

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

4276 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,e^2=g,f^2=g,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, 8735143);