#8115489

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
prime radical	#42424	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022685);