#10143698

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

3831 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,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, 10050894);