#8456836

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

4585 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)(3,8)(4,6)(5,7)(9,10,15,14)(11,18,13,16)(12,20,22,17)(19,23,21,24)(25,26,31,35)(27,32,38,36)(28,30,34,40)(29,39,37,33);(3,4)(6,8)(9,11,15,13)(10,16,14,18)(12,19,22,21)(17,23,20,24)(25,26,31,35)(27,36,38,32)(28,30,34,40)(29,33,37,39);(1,3,5,4)(2,6,7,8)(25,27,31,38)(26,32,35,36)(28,37,34,29)(30,33,40,39);(1,4,5,3)(2,6,7,8)(9,12,15,22)(10,17,14,20)(11,19,13,21)(16,23,18,24)(25,28)(26,30)(27,37)(29,38)(31,34)(32,33)(35,40)(36,39);(9,13,15,11)(10,18,14,16)(12,21,22,19)(17,24,20,23)(25,29,31,37)(26,33,35,39)(27,28,38,34)(30,32,40,36);(1,2)(3,6)(4,8)(5,7)(9,14,15,10)(11,18,13,16)(12,17,22,20)(19,23,21,24)(25,30)(26,34)(27,33)(28,35)(29,32)(31,40)(36,37)(38,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=i,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, 8364032);