#6438990

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#35659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3198 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)(9,10,16,22)(11,23,27,34)(12,28,30,36)(13,19,33,40)(14,20,15,21)(17,25,35,26)(18,29,37,24)(31,38,32,39);(5,6)(7,8)(9,11,15,25,16,27,14,26)(10,17,20,34,22,35,21,23)(12,13,24,31,30,33,29,32)(18,19,28,39,37,40,36,38);(2,4)(6,8)(9,12,14,29,16,30,15,24)(10,18,21,36,22,37,20,28)(11,13,26,32,27,33,25,31)(17,19,23,38,35,40,34,39);(6,8)(9,13)(10,19)(11,24)(12,26)(14,32)(15,31)(16,33)(17,28)(18,23)(20,39)(21,38)(22,40)(25,30)(27,29)(34,37)(35,36);(9,14,16,15)(10,20,22,21)(11,25,27,26)(12,29,30,24)(13,31,33,32)(17,23,35,34)(18,28,37,36)(19,38,40,39)

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^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, 6346186);