#9206483

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	#47895	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3973 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)(3,12)(4,6)(5,10)(7,9)(8,14)(11,15)(13,16)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,34,38,39)(35,41,44,40)(36,37,45,46)(42,47,48,43);(1,3,7,14)(2,8,9,12)(4,13,11,10)(5,15,16,6)(17,19)(18,22)(20,29)(21,30)(23,26)(24,25)(27,28)(31,32)(34,39)(36,45)(40,41)(42,48);(2,9)(4,11)(8,12)(10,13)(17,20,23,30)(18,24,27,31)(19,29,26,21)(22,25,28,32)(33,34)(35,40)(36,37)(38,39)(41,44)(42,43)(45,46)(47,48);(1,4,7,11)(2,6,9,15)(3,13,14,10)(5,12,16,8)(17,20)(18,24)(19,29)(21,26)(22,25)(23,30)(27,31)(28,32)(33,35,38,44)(34,40,39,41)(36,42,45,48)(37,43,46,47);(1,5,7,16)(2,10,9,13)(3,6,14,15)(4,8,11,12)(17,21,23,29)(18,25,27,32)(19,30,26,20)(22,24,28,31)(33,36)(34,37)(35,42)(38,45)(39,46)(40,43)(41,47)(44,48);(1,6)(2,11)(3,5)(4,9)(7,15)(8,10)(12,13)(14,16)(17,22,23,28)(18,26,27,19)(20,25,30,32)(21,31,29,24)(33,37)(34,36)(35,43)(38,46)(39,45)(40,42)(41,48)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=h,e^2=gi,f^2=gh,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, 9113679);