#6439014

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346210);