#7655203

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	#38308	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4375 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=g,e^2=gh,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, 7562399);