#10143607

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

4241 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=ghi,e^2=gh,f^2=h,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, 10050803);