#7732403

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

4130 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)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(26,30)(28,36)(31,33)(34,40);(1,3,5,8)(2,6,4,7)(11,17)(13,21)(15,19)(16,20);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(9,13,14,21)(10,16,12,20)(11,18,17,23)(15,22,19,24)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=h,e^2=hi,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, 7639599);