#8096860

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

4605 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,e^2=i,f^2=i,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, 8004056);