#438344

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345540);