#10143734

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

3844 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)(19,26)(20,29)(21,28)(22,24)(23,30)(25,31)(27,32)(33,34)(35,45)(36,42)(37,38)(39,43)(40,44)(41,47)(46,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,19,22,30)(18,23,24,26)(20,28,31,27)(21,29,32,25)(33,35)(34,41)(36,39)(37,46)(38,43)(40,47)(42,48)(44,45);(2,8)(3,11)(9,16)(10,12)(18,24)(20,31)(23,26)(27,28)(33,36,40,37)(34,42,44,38)(35,39,47,46)(41,48,45,43);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,32)(27,30)(33,37,40,36)(34,42,44,38)(35,46,47,39)(41,48,45,43);(1,4,6,14)(2,9,8,16)(3,12,11,10)(5,7,15,13)(17,19)(18,26)(20,27)(21,25)(22,30)(23,24)(28,31)(29,32)(33,38,40,42)(34,37,44,36)(35,43,47,48)(39,41,46,45);(1,5,6,15)(2,10,8,12)(3,9,11,16)(4,13,14,7)(17,21)(18,27)(19,29)(20,23)(22,32)(24,28)(25,30)(26,31)(33,39,40,46)(34,43,44,48)(35,36,47,37)(38,45,42,41)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=ghi,e^2=gh,f^2=gh,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, 10050930);