#415201

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

763 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)(11,19)(12,23)(13,26)(14,18)(15,21)(16,24)(17,29)(20,36)(22,31)(25,35)(27,34)(28,33)(30,39)(32,38)(37,40);(9,11,13,27,14,21,29,22)(10,15,17,34,18,19,26,31)(12,20,24,39,25,37,28,38)(16,32,23,40,33,30,35,36);(9,12)(10,16)(11,20,21,37)(13,28)(14,25)(15,30,19,32)(17,35)(18,33)(22,39,27,38)(23,26)(24,29)(31,36,34,40);(1,3,4,8)(2,5,6,7)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322397);