#8096319

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4236 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,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, 8003515);