#296401

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

778 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 203597);