#8096092

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	#35715	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4394 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=hi,f^2=hi,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, 8003288);