#432119

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

983 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=f,d^2=fi,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, 339315);