#6444381

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#35329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2794 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:

(2,4)(9,10,14,20,16,22,15,21)(11,23,26,35,28,34,27,17)(12,18,13,19,29,36,30,37)(24,39,40,38,31,33,25,32);(5,6)(7,8)(9,11,16,28)(10,17,22,35)(12,25,29,40)(13,31,30,24)(14,27,15,26)(18,33,36,39)(19,38,37,32)(20,34,21,23);(1,2,3,4)(6,8)(9,12,15,30,16,29,14,13)(10,18,21,37,22,36,20,19)(11,24,27,25,28,31,26,40)(17,32,34,33,35,38,23,39);(6,8)(9,13,14,29,16,30,15,12)(10,19,20,36,22,37,21,18)(11,25,26,24,28,40,27,31)(17,33,23,32,35,39,34,38);(12,29)(13,30)(18,36)(19,37)(24,31)(25,40)(32,38)(33,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,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, 6351577);