#424644

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

1227 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)(9,10,15,21)(11,23,26,36)(12,27,13,18)(14,22,16,20)(17,32,35,25)(19,30,38,29)(24,34,40,37)(28,33,31,39);(9,11)(10,17)(12,28)(13,31)(14,32)(15,26)(16,25)(18,37)(19,33)(20,36)(21,35)(22,23)(24,30)(27,34)(29,40)(38,39);(1,3,4,8)(2,5,6,7)(9,12,15,13)(10,18,21,27)(11,24,26,40)(14,29,16,30)(17,33,35,39)(19,22,38,20)(23,37,36,34)(25,31,32,28);(9,13,15,12)(10,19,21,38)(11,24,26,40)(14,29,16,30)(17,34,35,37)(18,22,27,20)(23,39,36,33)(25,28,32,31)

Polycyclic presentation:

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

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, 331840);