#9240949

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	#392	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)(20,21)(22,23)(24,30)(25,26)(28,29)(31,32)(33,34)(35,43)(36,37)(38,40)(39,42)(41,48)(44,45)(46,47);(3,11)(5,15)(7,16)(9,12)(18,23)(21,25)(24,27)(29,32)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,48,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,15)(10,16)(12,14)(17,19,22,30)(18,24,23,27)(20,31,26,28)(21,29,25,32)(33,36,39,41)(34,37,42,48)(35,44,38,46)(40,47,43,45);(17,20,22,26)(18,25,23,21)(19,28,30,31)(24,29,27,32)(33,37)(34,41)(35,45)(36,42)(38,47)(39,48)(40,44)(43,46);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,18)(19,27)(20,25)(21,26)(22,23)(24,30)(28,32)(29,31)(33,38,39,35)(34,40,42,43)(36,46,41,44)(37,47,48,45);(1,5)(2,9)(3,13)(4,7)(6,15)(8,11)(10,12)(14,16)(17,21,22,25)(18,26,23,20)(19,29,30,32)(24,28,27,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=gh,f^2=h,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, 9148145);