#6443228

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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

5096 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)(17,28)(22,30)(24,31)(25,32)(27,34)(29,33)(35,38)(36,39)(37,40);(1,2,3,4)(5,6)(7,8)(9,11,13,23,15,24,14,25)(10,16,19,21,20,32,18,31)(12,22,27,36,29,37,28,38)(17,30,33,35,34,40,26,39);(6,8)(9,12,14,28,15,29,13,27)(10,17,18,26,20,34,19,33)(11,22,25,38,24,37,23,36)(16,30,31,39,32,40,21,35);(6,8)(9,13,15,14)(10,18,20,19)(11,23,24,25)(12,27,29,28)(16,31,32,21)(17,26,34,33)(22,36,37,38)(30,39,40,35);(11,24)(16,32)(21,31)(22,37)(23,25)(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^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315547, #72315560, #72315579, …
maximal normal subgroup of	#72315547, #72315560, #72315579, …
maximal quotient of	#72315745, #72315860

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350424);