#8800149

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

4489 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=hi,e^2=hi,f^2=h,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, 8707345);