#8097138

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

4444 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004334);