#9910286

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

3880 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817482);