#8097087

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

4505 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=gh,f^2=g,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, 8004283);