#254413

order	512
nilpotency class	3
isoclinism	isoclinic class of #254371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3496	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

903 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:

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

Polycyclic presentation:

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

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, 161609);