#8456981

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

4499 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=gh,e^2=ghi,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, 8364177);