#8924691

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

4651 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831887);