#8699196

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

4561 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=gi,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, 8606392);