#8712393

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

4583 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8619589);