#9203811

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

4043 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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 9111007);