#9206767

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	#41730	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=gh,e^2=gi,f^2=g,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, 9113963);