#6438699

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂×#3010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3210 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)(6,7)(9,10,15,19)(11,16,23,32)(12,17,26,36)(13,27,14,18)(20,29,22,31)(21,30,38,39)(24,33,37,40)(25,35,28,34);(5,6,8,7)(9,11)(10,16)(12,24)(13,22)(14,20)(15,23)(17,33)(18,29)(19,32)(21,25)(26,37)(27,31)(28,38)(30,35)(34,39)(36,40);(2,4)(5,7,8,6)(11,20)(13,14)(16,29)(18,27)(21,24)(22,23)(25,28)(30,33)(31,32)(34,35)(37,38)(39,40);(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,30)(18,34)(19,36)(20,24)(22,37)(23,38)(27,35)(29,33)(31,40)(32,39);(9,13,15,14)(10,18,19,27)(11,22,23,20)(12,25,26,28)(16,29,32,31)(17,34,36,35)(21,37,38,24)(30,33,39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

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, 6345895);