#347893

order	512
nilpotency class	3
isoclinism	isoclinic class of #347878
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

1020 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)(18,28)(19,29)(21,36)(22,23)(24,25)(27,31)(30,39)(33,34)(35,37);(9,10,16,29,14,19,13,15)(11,17,20,31,23,32,22,27)(12,18,25,33,26,34,24,28)(21,30,35,40,37,39,36,38);(2,5)(3,8)(6,7)(9,11)(10,17)(12,21)(13,22)(14,23)(15,27)(16,20)(18,30)(19,32)(24,36)(25,35)(26,37)(28,38)(29,31)(33,40)(34,39);(9,12)(10,18)(11,21)(13,24)(14,26)(15,28)(16,25)(17,30)(19,34)(20,35)(22,36)(23,37)(27,38)(29,33)(31,40)(32,39)

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,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, 255089);