#10196082

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

4137 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)(3,8)(4,10)(5,11)(6,15)(7,9)(12,13)(14,16)(17,18)(19,27)(20,28)(21,25)(22,30)(23,24)(26,29)(31,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3,7,11)(2,8,9,5)(4,13,14,6)(10,12,16,15)(17,19,23,28)(18,20,24,27)(21,29,31,22)(25,30,32,26)(35,41)(38,45)(42,47)(46,48);(2,9)(3,11)(4,14)(12,15)(18,24)(20,27)(25,32)(26,30)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4)(2,10)(3,13)(5,15)(6,11)(7,14)(8,12)(9,16)(17,20,23,27)(18,19,24,28)(21,30,31,26)(22,25,29,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5)(2,11)(3,9)(4,12)(6,16)(7,8)(10,13)(14,15)(17,21,23,31)(18,25,24,32)(19,22,28,29)(20,26,27,30)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,6,7,13)(2,12,9,15)(3,4,11,14)(5,10,8,16)(17,22)(18,26)(19,21)(20,25)(23,29)(24,30)(27,32)(28,31)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=g,e^2=g,f^2=i,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, 10103278);