#10216867

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

3965 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=hi,e^2=ghi,f^2=ghi,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, 10124063);