#9240961

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

3702 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=h,e^2=h,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, 9148157);