#10349090

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

3998 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,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, 10256286);