#10142165

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

4274 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,48,44,42);(2,7)(4,8)(9,13)(12,14)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,35,39,45)(34,40,37,43)(36,44,47,38)(41,42,46,48);(2,7)(3,10)(9,13)(11,16)(18,23)(19,25)(24,32)(26,27)(33,36,39,47)(34,41,37,46)(35,38,45,44)(40,48,43,42);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,42)(43,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,20,22,29)(18,24,23,32)(19,26,25,27)(21,30,31,28)(33,37,39,34)(35,40,45,43)(36,46,47,41)(38,48,44,42);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)(33,38,39,44)(34,42,37,48)(35,36,45,47)(40,46,43,41)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=hi,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, 10049361);