#8665313

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

4518 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8572509);