#10216663

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

3899 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)(18,21)(19,25)(24,28)(26,29)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,22,21)(19,23,25,27)(20,28,30,24)(26,32,29,31)(34,40)(35,37)(42,46)(43,45);(17,19,22,25)(18,23,21,27)(20,26,30,29)(24,31,28,32)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,22,18)(19,27,25,23)(20,24,30,28)(26,31,29,32)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=i,e^2=ghi,f^2=gi,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, 10123859);