#6438040

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#3485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4744 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)(9,10,13,19,15,21,14,20)(11,22,24,33,26,32,25,16)(12,18,28,17,29,34,27,35)(23,37,39,36,40,30,38,31);(2,4)(5,6)(7,8)(9,11,15,26)(10,16,21,33)(12,23,29,40)(13,25,14,24)(17,30,35,37)(18,31,34,36)(19,32,20,22)(27,39,28,38);(6,8)(9,12)(10,17)(11,23)(13,27)(14,28)(15,29)(16,30)(18,19)(20,34)(21,35)(22,36)(24,38)(25,39)(26,40)(31,32)(33,37);(6,8)(9,12)(10,18)(11,23)(13,28)(14,27)(15,29)(16,31)(17,19)(20,35)(21,34)(22,37)(24,39)(25,38)(26,40)(30,32)(33,36);(9,13,15,14)(10,19,21,20)(11,24,26,25)(12,27,29,28)(16,22,33,32)(17,18,35,34)(23,38,40,39)(30,36,37,31)

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,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,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, 6345236);