#415281

order	512
nilpotency class	3
isoclinism	isoclinic class of #397240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
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

962 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)(9,10,14,28,13,17,27,18)(11,19,33,34,22,15,21,30)(12,23,26,16,25,38,36,32)(20,35,40,29,24,39,37,31);(9,11,14,21,13,22,27,33)(10,15,18,30,17,19,28,34)(12,24,26,40,25,20,36,37)(16,31,23,35,32,29,38,39);(9,12,13,25)(10,16)(11,20)(14,26,27,36)(15,29,19,31)(17,32)(18,23)(21,37)(22,24)(28,38)(30,39,34,35)(33,40);(1,3,4,8)(2,5,6,7)(10,17)(15,19)(16,32)(18,28)(23,38)(29,31)(30,34)(35,39)

Polycyclic presentation:

2^9|a^2=efh,b^2=f,c^2=e,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,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, 322477);