#403036

order	512
nilpotency class	3
isoclinism	isoclinic class of #396101
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1316 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 310232);