#10349091

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

4189 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)(3,7,11,9)(5,12,13,8)(10,15,16,14)(17,18)(19,25)(20,22)(21,24)(23,27)(26,31)(28,29)(30,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,46)(42,47)(45,48);(1,2,6,4)(3,9,11,7)(5,12,13,8)(10,14,16,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(2,4)(7,9)(8,12)(14,15)(21,29)(24,28)(26,32)(30,31)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20,22,18)(19,23,27,25)(21,24,29,28)(26,31,32,30)(33,35)(34,38)(36,42)(37,43)(39,47)(40,41)(44,48)(45,46);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,46)(45,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,19)(18,25)(20,23)(21,26)(22,27)(24,31)(28,30)(29,32)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10256287);