#9913267

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

4348 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=g,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, 9820463);