#7655541

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=i,e^2=h,f^2=h,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, 7562737);