#8458678

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

4464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^2=i,f^2=gi,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, 8365874);