#6443242

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^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^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315547, #72315582, #72316094, …
maximal normal subgroup of	#72315547, #72315582, #72316094, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350438);