#9508915

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	#56854	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=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, 9416111);