#8463373

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

4433 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,e^2=gi,f^2=hi,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, 8370569);