#7656695

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

4139 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)(9,10)(11,22)(12,19)(13,18)(14,20)(15,21)(16,23)(17,24)(26,28)(27,33)(32,36)(35,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,23)(10,16,21,22)(12,20,24,13)(14,19,18,17)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(9,12)(10,17)(11,20)(13,23)(14,16)(15,24)(18,22)(19,21)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,3,5,4)(2,7,6,8)(9,13,15,20)(10,18,21,14)(11,12,23,24)(16,17,22,19)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,4,5,3)(2,8,6,7)(9,12)(10,19)(11,20)(13,23)(14,22)(15,24)(16,18)(17,21)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40);(9,14,15,18)(10,20,21,13)(11,19,23,17)(12,16,24,22)

Polycyclic presentation:

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

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, 7563891);