#8445583

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

4460 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,e^2=i,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, 8352779);