#7732044

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,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, 7639240);