#415526

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

693 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322722);