#9243503

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

3999 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9150699);