#8943975

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

4237 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851171);