#8464276

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

4434 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=gi,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, 8371472);