#8772854

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

4651 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8680050);