#10464580

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10371776);