#7642936

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	#39165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4390 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)(10,15)(13,21)(16,22)(17,23)(25,26)(27,32)(28,30)(29,37)(31,34)(33,39)(35,38)(36,40);(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,16,19,22)(12,17,20,23)(13,18,21,24)(25,27,31,28)(26,32,34,30)(29,35,39,36)(33,40,37,38);(10,15)(12,20)(16,22)(18,24)(25,28,31,27)(26,32,34,30)(29,36,39,35)(33,40,37,38);(1,3,5,4)(2,7,6,8)(9,11,14,19)(10,16,15,22)(12,18,20,24)(13,17,21,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37);(9,12,14,20)(10,17,15,23)(11,18,19,24)(13,16,21,22)(25,30,31,32)(26,28,34,27)(29,38,39,40)(33,36,37,35);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,18,15,24)(11,17,19,23)(12,16,20,22)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=g,e^2=hi,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, 7550132);