#8325977

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

4390 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8233173);