#8096305

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	#45606	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4323 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)(3,4)(5,7)(6,8)(9,10,15,18)(11,13,20,16)(12,21,14,17)(19,24,22,23);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,22)(13,18)(14,19)(15,20)(17,23)(21,24)(26,30)(31,37)(32,38)(39,40);(9,12,15,14)(10,17,18,21)(11,19,20,22)(13,23,16,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,23)(14,24)(15,16)(17,19)(18,20)(21,22);(1,4,5,6)(2,3,7,8)(9,11)(10,13)(12,19)(14,22)(15,20)(16,18)(17,23)(21,24)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(1,3)(2,4)(5,8)(6,7)(9,14,15,12)(10,17,18,21)(11,19,20,22)(13,24,16,23)(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=h,e^2=gi,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, 8003501);