#9015580

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922776);