#8969527

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

4351 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=gh,f^2=g,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, 8876723);