#8587877

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

4537 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8495073);