#347923

order	512
nilpotency class	3
isoclinism	isoclinic class of #347908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1016 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255119);