#8923662

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

4151 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)(33,34,38,40)(35,39,43,47)(36,41,42,37)(44,45,46,48);(2,7)(4,9)(8,10)(12,14)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35)(34,39)(36,46)(37,45)(38,43)(40,47)(41,48)(42,44);(2,7)(3,11)(5,15)(12,14)(18,19)(21,29)(23,25)(27,32)(34,40)(35,43)(37,41)(44,46);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,35)(34,39)(36,46)(37,45)(38,43)(40,47)(41,48)(42,44);(1,4,6,9)(2,5,7,15)(3,12,11,14)(8,16,10,13)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,38,42)(34,41,40,37)(35,44,43,46)(39,45,47,48);(1,5)(2,9)(3,13)(4,7)(6,15)(8,12)(10,14)(11,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,41)(34,42,40,36)(35,45,43,48)(39,44,47,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=ghi,e^2=gi,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, 8830858);