#9240983

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

4015 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9148179);