#9016605

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

4081 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)(7,10)(8,13)(11,14)(15,16)(17,18,23,21)(19,24,22,29)(20,25,27,30)(26,31,28,32);(2,6)(7,9)(8,12)(14,15)(17,19,23,22)(18,24,21,29)(20,26,27,28)(25,31,30,32)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(2,6)(3,10)(4,13)(14,15)(18,21)(19,22)(20,27)(31,32)(33,35,39,46)(34,40,43,44)(36,37,42,45)(38,47,48,41);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20)(18,25)(19,26)(21,30)(22,28)(23,27)(24,31)(29,32)(33,36,39,42)(34,38,43,48)(35,45,46,37)(40,41,44,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21,23,18)(19,24,22,29)(20,25,27,30)(26,32,28,31)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,19)(18,24,21,29)(20,28,27,26)(25,31,30,32)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=ghi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8923801);