#415110

order	512
nilpotency class	3
isoclinism	isoclinic class of #395781
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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

873 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)(3,7)(4,5)(6,8)(9,10,13,18,15,19,14,30)(11,20,22,32,24,36,23,16)(12,25,28,35,29,34,27,17)(21,31,26,40,38,39,37,33);(9,11,15,24)(10,16,19,32)(12,26,29,37)(13,23,14,22)(17,33,35,40)(18,36,30,20)(21,27,38,28)(25,39,34,31);(2,5)(6,7)(9,12,13,27,15,29,14,28)(10,17,18,34,19,35,30,25)(11,21,22,37,24,38,23,26)(16,31,20,33,32,39,36,40);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

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, 322306);