#9206463

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	#48185	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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)(3,8)(4,7)(5,9)(6,11)(10,16)(12,15)(13,14)(17,18)(19,25)(20,24)(21,23)(22,26)(27,32)(28,31)(29,30)(33,34,39,37)(35,36,42,40)(38,45,46,41)(43,47,48,44);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,13)(15,16)(17,19,22,21)(18,23,26,25)(20,28,30,27)(24,32,29,31)(34,37)(35,42)(41,45)(43,48);(4,12)(7,15)(10,13)(14,16)(17,18,22,26)(19,23,21,25)(20,24,30,29)(27,31,28,32)(33,35,39,42)(34,40,37,36)(38,43,46,48)(41,47,45,44);(1,4,5,12)(2,7,9,15)(3,10,11,13)(6,14,8,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,30)(27,32)(28,31)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,48)(46,47);(1,3)(2,8)(4,13)(5,11)(6,9)(7,14)(10,12)(15,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,32)(26,29)(33,37,39,34)(35,36,42,40)(38,45,46,41)(43,44,48,47);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,15)(13,16)(17,21,22,19)(18,25,26,23)(20,27,30,28)(24,31,29,32)(33,38,39,46)(34,41,37,45)(35,43,42,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113659);