#413070

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1552 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320266);