#8422070

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=h,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, 8329266);