#8119927

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

4553 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,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, 8027123);