#424731

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1085 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331927);