#8993978

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

4050 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)(3,9,10,7)(5,12,13,8)(11,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(2,4)(7,9)(8,12)(14,15)(17,19,22,25)(18,23,21,27)(20,26,28,29)(24,31,30,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(2,4)(3,10)(5,13)(14,15)(18,21)(19,25)(20,28)(31,32)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,29)(27,32)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,4,6,2)(3,7,10,9)(5,12,13,8)(11,14,16,15)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,28)(30,31)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,21,22,18)(19,27,25,23)(20,30,28,24)(26,32,29,31)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=hi,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, 8901174);