#6373200

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2035 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280396);