#9935920

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

4218 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,24)(20,25)(21,29)(22,23)(26,31)(27,28)(30,32)(33,34,39,41)(35,45,46,40)(36,37,43,42)(38,48,47,44);(3,9)(7,13)(10,15)(14,16)(18,23)(19,25)(26,29)(27,32)(33,35)(34,40)(36,47)(37,48)(38,43)(39,46)(41,45)(42,44);(2,6)(7,13)(8,11)(14,16)(18,23)(19,25)(26,29)(27,32)(34,41)(35,46)(36,43)(44,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,25)(18,24,23,20)(21,27,31,32)(26,30,29,28)(33,36,39,43)(34,42,41,37)(35,47,46,38)(40,44,45,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20,22,24)(18,25,23,19)(21,30,31,28)(26,27,29,32)(33,37,39,42)(34,43,41,36)(35,44,46,48)(38,40,47,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,31)(18,26,23,29)(19,27,25,32)(20,28,24,30)(33,38,39,47)(34,44,41,48)(35,43,46,36)(37,40,42,45)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9843116);