#10045695

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

4027 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,34,39,40)(35,41,45,48)(36,42,46,37)(38,43,47,44);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(19,20)(23,26)(27,28)(31,32)(34,40)(36,46)(41,48)(43,44);(2,8)(4,9)(7,14)(10,15)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,35,39,45)(34,41,40,48)(36,44,46,43)(37,47,42,38);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,36)(34,42)(35,44)(37,40)(38,41)(39,46)(43,45)(47,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,37,39,42)(34,36,40,46)(35,38,45,47)(41,43,48,44);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38)(34,43)(35,37)(36,41)(39,47)(40,44)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=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, 9952891);