#8993565

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

4212 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=ghi,e^2=g,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, 8900761);