#6438026

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×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4700 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)(9,10,13,17,15,20,14,19)(11,21,23,31,25,29,24,16)(12,18,27,32,26,33,28,34)(22,35,37,40,36,39,38,30);(2,4)(5,6)(7,8)(9,11,15,25)(10,16,20,31)(12,22,26,36)(13,24,14,23)(17,29,19,21)(18,30,33,40)(27,38,28,37)(32,39,34,35);(6,8)(10,17)(13,14)(16,29)(18,32)(19,20)(21,31)(23,24)(27,28)(30,39)(33,34)(35,40)(37,38);(6,8)(9,12)(10,18)(11,22)(13,27)(14,28)(15,26)(16,30)(17,32)(19,34)(20,33)(21,35)(23,37)(24,38)(25,36)(29,39)(31,40);(12,26)(18,33)(22,36)(27,28)(30,40)(32,34)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,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, 6345222);