#9059308

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^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, 8966504);