#398941

order	512
nilpotency class	3
isoclinism	isoclinic class of #395252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

917 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:

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

Polycyclic presentation:

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

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, 306137);