#8436625

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

4453 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)(3,4)(5,7)(6,8)(9,10,15,16)(11,20,21,17)(12,24,18,13)(14,19,22,23)(25,26,31,30)(27,33,28,32)(29,37,38,34)(35,39,36,40);(1,3)(2,6)(4,7)(5,8)(10,16)(11,21)(12,18)(19,23)(25,27,31,28)(26,32,30,33)(29,36,38,35)(34,40,37,39);(9,11)(10,17)(12,22)(13,23)(14,18)(15,21)(16,20)(19,24)(25,28,31,27)(26,33,30,32)(29,36,38,35)(34,40,37,39);(2,7)(4,6)(9,12,15,18)(10,13,16,24)(11,22,21,14)(17,23,20,19)(25,29,31,38)(26,34,30,37)(27,35,28,36)(32,39,33,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,18)(11,23)(12,16)(14,17)(15,24)(19,21)(20,22);(1,4,5,6)(2,3,7,8)(9,14,15,22)(10,19,16,23)(11,18,21,12)(13,17,24,20)(25,30,31,26)(27,33,28,32)(29,34,38,37)(35,39,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,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, 8343821);