#10143256

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18)(19,26)(20,28)(21,23)(22,25)(24,29)(27,31)(30,32)(33,34,39,44)(35,43,38,40)(36,46,47,41)(37,42,48,45);(1,3,7,4)(2,8,9,10)(5,12,15,13)(6,11,16,14)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32)(33,35,39,38)(34,40,44,43)(36,45,47,42)(37,46,48,41);(2,9)(3,4)(11,14)(12,13)(33,36,39,47)(34,41,44,46)(35,45,38,42)(37,43,48,40);(1,4,7,3)(2,10,9,8)(5,13,15,12)(6,14,16,11)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32)(33,36)(34,41)(35,45)(37,40)(38,42)(39,47)(43,48)(44,46);(1,5,7,15)(2,11,9,14)(3,13,4,12)(6,10,16,8)(17,21,22,19)(18,23,25,26)(20,27,29,30)(24,32,28,31)(33,37)(34,42)(35,41)(36,43)(38,46)(39,48)(40,47)(44,45);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,8,15,10)(17,18)(19,23)(20,24)(21,26)(22,25)(27,31)(28,29)(30,32)(33,38,39,35)(34,43,44,40)(36,42,47,45)(37,41,48,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=i,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, 10050452);