#9017338

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

4162 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924534);