#8993661

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

4054 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=ghi,e^2=g,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, 8900857);