#6439008

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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

3178 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)(11,18,25,31)(12,17,24,32)(13,19,29,36)(14,20,15,21)(23,33,38,39)(26,30,37,40)(27,34,28,35);(5,6)(7,8)(9,11,15,24,16,25,14,12)(10,17,20,31,22,32,21,18)(13,23,27,37,29,38,28,26)(19,30,35,39,36,40,34,33);(2,4)(6,8)(9,12,14,25,16,24,15,11)(10,18,21,32,22,31,20,17)(13,26,28,38,29,37,27,23)(19,33,34,40,36,39,35,30);(6,8)(9,13)(10,19)(11,23)(12,26)(14,28)(15,27)(16,29)(17,30)(18,33)(20,35)(21,34)(22,36)(24,37)(25,38)(31,39)(32,40);(9,14,16,15)(10,20,22,21)(11,24,25,12)(13,27,29,28)(17,18,32,31)(19,34,36,35)(23,26,38,37)(30,39,40,33)

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=dh,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346204);