#10147565

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

3855 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,14)(13,15)(17,18)(19,25)(20,30)(21,23)(22,26)(24,27)(28,31)(29,32)(33,34,39,37)(35,43,45,40)(36,44,46,41)(38,42,48,47);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(17,19,24,21)(18,25,27,23)(20,29,22,28)(26,31,30,32)(33,35)(34,40)(36,42)(37,43)(38,44)(39,45)(41,48)(46,47);(2,4)(8,10)(9,12)(15,16)(17,20,24,22)(18,26,27,30)(19,28,21,29)(23,31,25,32)(33,34,39,37)(35,40,45,43)(36,41,46,44)(38,42,48,47);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,21,24,19)(18,23,27,25)(20,28,22,29)(26,32,30,31)(33,36)(34,41)(35,42)(37,44)(38,43)(39,46)(40,48)(45,47);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,22,24,20)(18,26,27,30)(19,29,21,28)(23,31,25,32)(33,37,39,34)(35,43,45,40)(36,41,46,44)(38,42,48,47);(1,6,7,3)(2,10,4,8)(5,11,14,13)(9,15,12,16)(17,23)(18,19)(20,31)(21,27)(22,32)(24,25)(26,28)(29,30)(33,38,39,48)(34,42,37,47)(35,44,45,41)(36,43,46,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=h,e^2=gi,f^2=g,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, 10054761);