#8096919

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

4442 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)(3,7)(4,5)(6,8)(9,10,15,16)(11,18,12,17)(13,19,23,22)(14,20,24,21)(25,26,31,29)(27,34,32,28)(30,33,38,37)(35,40,39,36);(2,4)(6,7)(10,16)(11,12)(19,22)(20,21)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,17,16,18)(13,21,23,20)(14,19,24,22)(25,26,31,29)(27,28,32,34)(30,33,38,37)(35,36,39,40);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,18,16,17)(13,20,23,21)(14,22,24,19)(25,28,31,34)(26,32,29,27)(30,36,38,40)(33,39,37,35);(9,13)(10,19)(11,21)(12,20)(14,18)(15,23)(16,22)(17,24)(25,29,31,26)(27,34,32,28)(30,33,38,37)(35,36,39,40);(9,14)(10,20)(11,19)(12,22)(13,17)(15,24)(16,21)(18,23)(25,30,31,38)(26,33,29,37)(27,35,32,39)(28,36,34,40)

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=g,e^2=gh,f^2=gh,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, 8004115);