#8889415

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

4092 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,22,24)(19,26,29,23)(20,30,25,21)(27,32,31,28)(34,36)(37,43)(41,44)(45,48);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,19,22,29)(18,23,24,26)(20,27,25,31)(21,32,30,28)(33,34,39,36)(35,37,40,43)(38,44,46,41)(42,48,47,45);(2,6)(7,9)(8,12)(14,15)(18,24)(20,25)(23,26)(27,31)(33,35,39,40)(34,37,36,43)(38,42,46,47)(41,45,44,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,22,25)(18,21,24,30)(19,27,29,31)(23,32,26,28)(33,36,39,34)(35,37,40,43)(38,44,46,41)(42,45,47,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,19)(18,23)(20,31)(21,28)(22,29)(24,26)(25,27)(30,32)(33,37)(34,40)(35,36)(38,45)(39,43)(41,47)(42,44)(46,48);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21)(18,25)(19,28)(20,24)(22,30)(23,27)(26,31)(29,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=g,e^2=i,f^2=i,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, 8796611);