#8691875

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

4628 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)(9,10,15,19)(11,16,20,22)(12,17,21,24)(13,14,18,23)(25,26,30,33)(27,31,35,40)(28,36,29,32)(34,39,37,38);(1,3)(2,4)(5,7)(6,8)(11,20)(12,21)(16,22)(17,24)(25,27)(26,31)(28,37)(29,34)(30,35)(32,39)(33,40)(36,38);(9,11)(10,16)(12,13)(14,17)(15,20)(18,21)(19,22)(23,24)(25,28,30,29)(26,32,33,36)(27,34,35,37)(31,38,40,39);(2,5)(3,8)(9,12)(10,17)(11,13)(14,16)(15,21)(18,20)(19,24)(22,23)(25,27)(26,31)(28,34)(29,37)(30,35)(32,38)(33,40)(36,39);(1,4)(2,3)(5,8)(6,7)(9,13)(10,14)(11,12)(15,18)(16,17)(19,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,36)(30,33)(34,38)(35,40)(37,39);(1,5,6,2)(3,7,8,4)(9,14)(10,18)(11,17)(12,22)(13,19)(15,23)(16,21)(20,24)(25,29,30,28)(26,32,33,36)(27,34,35,37)(31,39,40,38)

Polycyclic presentation:

2^9|a^2=gi,c^2=h,f^2=gh,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, 8599071);