#8455921

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

4489 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=g,e^2=i,f^2=g,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, 8363117);