#8095798

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

4311 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=hi,e^2=ghi,f^2=gh,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, 8002994);