#9203810

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

4232 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9111006);