#7642120

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	#37093	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549316);