#7645926

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

4374 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553122);