#6438962

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3250 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:

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346158);