#7654913

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4264 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562109);