#9267021

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	#54498	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3858 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)(4,5)(6,16)(7,9)(8,13)(10,15)(11,14)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(1,3,7,13)(2,8,9,12)(4,14,15,6)(5,16,10,11)(17,18,23,25)(19,26,20,21)(22,30,31,24)(27,29,28,32)(34,39)(35,41)(40,46)(42,47);(2,9)(3,13)(4,15)(11,16)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,30,32)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,4)(2,10)(3,14)(5,9)(6,13)(7,15)(8,11)(12,16)(17,20,23,19)(18,21,25,26)(22,28,31,27)(24,29,30,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,5,7,10)(2,4,9,15)(3,11,13,16)(6,12,14,8)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,30)(31,32);(1,6,7,14)(2,11,9,16)(3,4,13,15)(5,12,10,8)(17,22)(18,24)(19,27)(20,28)(21,29)(23,31)(25,30)(26,32)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=gi,e^2=h,f^2=g,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, 9174217);