#9851047

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

4133 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)(3,10)(4,5)(6,8)(7,14)(9,11)(12,15)(13,16)(17,18)(19,24)(20,25)(21,29)(22,23)(26,31)(27,28)(30,32)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,3)(2,7)(4,12)(5,13)(6,14)(8,10)(9,15)(11,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,32)(29,31)(34,40)(35,37)(42,46)(43,45);(2,8)(4,11)(7,10)(12,16)(18,23)(19,25)(26,29)(27,32)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,4,6,11)(2,9,8,5)(3,12,14,16)(7,15,10,13)(17,19,22,25)(18,24,23,20)(21,27,31,32)(26,30,29,28)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,3)(2,10)(4,12)(5,15)(6,14)(7,8)(9,13)(11,16)(17,20,22,24)(18,25,23,19)(21,30,31,28)(26,27,29,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,5,6,9)(2,11,8,4)(3,13,14,15)(7,16,10,12)(17,21)(18,26)(19,27)(20,28)(22,31)(23,29)(24,30)(25,32)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9758243);