#10128128

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

4139 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10035324);