#8695991

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

4631 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603187);