#8445242

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

4449 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8352438);