#405859

order	512
nilpotency class	3
isoclinism	isoclinic class of #395775
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

990 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)(3,5)(4,6)(7,8)(9,10,14,19,15,20,13,18)(11,21,23,16,25,32,24,33)(12,26,28,34,29,35,30,17)(22,37,39,40,27,31,38,36);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,31)(18,32)(19,21)(20,33)(22,29)(26,40)(28,39)(30,38)(34,37)(35,36);(9,12,13,28,15,29,14,30)(10,17,18,26,20,34,19,35)(11,22,24,39,25,27,23,38)(16,31,21,36,33,37,32,40);(2,6)(5,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,29,30)(16,32,33,21)(17,26,34,35)(22,38,27,39)(31,40,37,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313055);