#7732344

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

4120 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)(9,10,15,12)(11,18,14,16)(13,20,22,17)(19,23,21,24)(25,26,30,28)(27,31,33,35)(29,32,37,36)(34,38,39,40);(1,3,4,8)(2,5,6,7)(11,14)(16,18)(19,21)(23,24)(27,33)(29,37)(31,35)(32,36);(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24);(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,28,30,26)(27,31,33,35)(29,36,37,32)(34,38,39,40);(9,14,15,11)(10,16,12,18)(13,21,22,19)(17,23,20,24)(25,29,30,37)(26,32,28,36)(27,34,33,39)(31,38,35,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

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, 7639540);