#6438811

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3162 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346007);