#7642577

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

4488 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)(10,15)(13,23)(16,22)(17,24)(25,26)(27,32)(28,30)(29,37)(31,34)(33,39)(35,38)(36,40);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,27,31,28)(26,32,34,30)(29,35,39,36)(33,40,37,38);(10,15)(12,21)(16,22)(18,19)(25,28,31,27)(26,32,34,30)(29,36,39,35)(33,40,37,38);(1,3,5,4)(2,7,6,8)(9,11,14,20)(10,16,15,22)(12,19,21,18)(13,17,23,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37);(9,12)(10,17)(11,19)(13,22)(14,21)(15,24)(16,23)(18,20)(25,30,31,32)(26,28,34,27)(29,38,39,40)(33,36,37,35);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,18,15,19)(11,17,20,24)(12,22,21,16)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)

Polycyclic presentation:

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

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, 7549773);