#6437864

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3489	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4676 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)(9,10,13,19,15,18,14,20)(11,21,23,31,25,32,24,16)(12,26,28,17,29,34,27,33)(22,36,38,35,39,30,37,40);(2,4)(5,6)(7,8)(9,11,15,25)(10,16,18,31)(12,22,29,39)(13,24,14,23)(17,30,33,36)(19,32,20,21)(26,40,34,35)(27,38,28,37);(6,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,33)(19,26)(20,34)(21,35)(23,37)(24,38)(25,39)(31,36)(32,40);(6,8)(10,18)(16,31)(17,33)(19,20)(21,32)(26,34)(30,36)(35,40);(9,13,15,14)(10,19,18,20)(11,23,25,24)(12,27,29,28)(16,21,31,32)(17,26,33,34)(22,37,39,38)(30,35,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345060);