#8818079

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

4392 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)(25,26,32,36)(27,29,30,33)(28,35,31,34)(37,38,40,39);(1,3,5,8)(2,6,7,4)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,27)(26,33)(28,37)(29,36)(30,32)(31,40)(34,38)(35,39);(1,3)(2,4)(5,8)(6,7)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,28,32,31)(26,34,36,35)(27,37,30,40)(29,39,33,38);(1,4)(2,3)(5,6)(7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,29,32,33)(26,27,36,30)(28,38,31,39)(34,40,35,37);(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,30,32,27)(26,33,36,29)(28,37,31,40)(34,39,35,38);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,31,32,28)(26,35,36,34)(27,37,30,40)(29,39,33,38)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725275);