#405789

order	512
nilpotency class	3
isoclinism	isoclinic class of #395409
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1848 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)(9,10,13,18,15,20,14,19)(11,21,23,16,24,31,28,32)(12,17,29,35,27,36,26,34)(22,37,25,30,39,33,38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,25)(13,23)(14,28)(15,24)(17,33)(18,31)(19,21)(20,32)(22,26)(27,38)(29,39)(30,34)(35,40)(36,37);(9,12)(10,17)(11,22)(13,26)(14,29)(15,27)(16,30)(18,34)(19,35)(20,36)(21,33)(23,38)(24,39)(25,28)(31,37)(32,40);(2,6)(5,8)(9,13,15,14)(10,18,20,19)(11,23,24,28)(12,26,27,29)(16,31,32,21)(17,34,36,35)(22,38,39,25)(30,37,40,33)

Polycyclic presentation:

2^9|a^2=ef,b^2=h,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=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, 312985);