#9057381

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

4121 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8964577);