#8095544

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

4595 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002740);