#7655322

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

4492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=g,e^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, 7562518);