#405783

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,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, 312979);