#6438033

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

Polycyclic presentation:

2^9|a^2=fi,c^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, 6345229);