#7648357

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4170 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=h,e^2=h,f^2=i,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, 7555553);