#9058460

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

4122 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965656);