#8900057

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

4470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8807253);