#7655436

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=i,e^2=i,f^2=i,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, 7562632);