#8425382

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

4695 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332578);