#301860

order	512
nilpotency class	3
isoclinism	isoclinic class of #301797
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1011 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=f,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gi,d^c=dhi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209056);