#296429

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	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

819 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)(11,17)(12,19)(14,20)(16,22)(18,23)(21,24)(25,26,29,32)(27,33,35,30)(28,31,37,39)(34,38,36,40);(2,5)(6,7)(9,10,13,15)(11,14,18,21)(12,16,19,22)(17,20,23,24)(25,27)(26,30)(28,36)(29,35)(31,38)(32,33)(34,37)(39,40);(9,11,12,17,13,18,19,23)(10,14,16,20,15,21,22,24)(25,28)(26,31)(27,34,35,36)(29,37)(30,38,33,40)(32,39);(1,3,4,8)(2,6,5,7)(10,15)(11,18)(16,22)(17,23)(26,32)(30,33)(31,39)(38,40)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=f,d^2=ei,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,d^c=dh,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, 203625);