#8093692

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

4641 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^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, 8000888);