#8093786

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

4316 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=h,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, 8000982);