#412116

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1350 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:

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

Polycyclic presentation:

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

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, 319312);