#7645794

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	#35130	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4316 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)(10,12)(11,17)(16,20)(19,23)(26,30)(27,33)(32,37)(34,39);(1,2)(3,6)(4,5)(7,8)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(26,30)(31,36)(32,37)(38,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=g,e^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, 7552990);