#8697522

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

4543 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:

(9,10,14,19)(11,15,21,24)(12,17,22,16)(13,20,23,18)(26,31)(32,33)(34,37)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,18)(13,17)(14,21)(16,23)(19,24)(20,22)(25,26,30,31)(27,32,28,33)(29,34,38,37)(35,39,36,40);(1,3,5,6)(2,4,7,8)(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,27,30,28)(26,32,31,33)(29,35,38,36)(34,39,37,40);(1,4,5,8)(2,3,7,6)(9,12)(10,17)(11,18)(13,15)(14,22)(16,19)(20,21)(23,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,40)(36,39);(1,2)(3,4)(5,7)(6,8)(9,13)(10,18)(11,17)(12,15)(14,23)(16,21)(19,20)(22,24)(25,28,30,27)(26,33,31,32)(29,35,38,36)(34,39,37,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,20)(13,16)(14,21)(17,23)(18,22)(19,24)(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=ghi,d^2=hi,e^2=g,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, 8604718);