#8119910

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

4283 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:

(2,4)(6,7)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,2,5,4)(3,7,8,6)(10,16)(11,12)(18,23)(19,20)(25,27,31,28)(26,32,35,33)(29,36,34,38)(30,37,40,39);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,27,31,28)(26,33,35,32)(29,38,34,36)(30,37,40,39);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,28,31,27)(26,33,35,32)(29,38,34,36)(30,39,40,37);(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,29,31,34)(26,30,35,40)(27,36,28,38)(32,37,33,39);(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(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=gh,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

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, 8027106);