#8096408

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003604);