#10142173

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

4067 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,15)(14,16)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28)(33,34)(35,43)(36,41)(37,46)(38,40)(39,42)(44,48)(45,47);(2,7)(4,14)(8,11)(10,13)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,35,39,38)(34,40,42,43)(36,44,37,45)(41,47,46,48);(2,7)(3,12)(9,16)(10,13)(18,23)(19,25)(24,32)(26,27)(33,36,39,37)(34,41,42,46)(35,44,38,45)(40,47,43,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,14)(15,16)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,37,39,36)(34,41,42,46)(35,45,38,44)(40,47,43,48);(1,4,6,14)(2,9,7,16)(3,13,12,10)(5,8,15,11)(17,20,22,29)(18,24,23,32)(19,26,25,27)(21,30,31,28)(33,34)(35,40)(36,41)(37,46)(38,43)(39,42)(44,47)(45,48);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)(33,38,39,35)(34,40,42,43)(36,45,37,44)(41,47,46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=ghi,e^2=h,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, 10049369);