#10143653

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

3853 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)(6,14)(7,9)(12,15)(13,16)(17,18,23,21)(19,27,29,24)(20,30,28,25)(22,26,32,31)(34,37)(35,41)(40,44)(42,45);(1,3,7,12)(2,8,9,15)(4,13,14,5)(6,11,10,16)(17,19,23,29)(18,24,21,27)(20,22,28,32)(25,26,30,31)(33,34,38,37)(35,39,41,43)(36,44,46,40)(42,48,45,47);(2,9)(3,12)(4,14)(11,16)(33,35,38,41)(34,39,37,43)(36,42,46,45)(40,47,44,48);(1,4)(2,10)(3,13)(5,12)(6,9)(7,14)(8,16)(11,15)(17,20,23,28)(18,25,21,30)(19,22,29,32)(24,26,27,31)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,5,7,13)(2,11,9,16)(3,14,12,4)(6,15,10,8)(17,21,23,18)(19,27,29,24)(20,25,28,30)(22,26,32,31)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,45)(43,48);(1,6,7,10)(2,4,9,14)(3,11,12,16)(5,8,13,15)(17,22)(18,26)(19,28)(20,29)(21,31)(23,32)(24,30)(25,27)(33,37,38,34)(35,43,41,39)(36,40,46,44)(42,47,45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,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, 10050849);