#10069240

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

4085 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)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(19,25)(23,26)(27,31)(30,32)(33,34)(35,42)(36,40)(37,45)(38,39)(41,46)(43,44)(47,48);(3,9)(7,13)(10,15)(14,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,35,38,40)(34,36,39,42)(37,43,46,47)(41,44,45,48);(2,6)(7,13)(8,11)(14,16)(18,20)(23,26)(24,28)(30,32)(34,39)(36,42)(41,45)(44,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36,38,42)(34,40,39,35)(37,44,46,48)(41,47,45,43);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,34)(35,42)(36,40)(37,45)(38,39)(41,46)(43,44)(47,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,46)(34,41,39,45)(35,43,40,47)(36,44,42,48)

Polycyclic presentation:

2^9|b^2=gi,d^2=gi,e^2=ghi,f^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, 9976436);