#347929

order	512
nilpotency class	3
isoclinism	isoclinic class of #347908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1027 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,18,14)(12,16,25,32)(13,26,29,17)(21,33,31,24)(22,28,27,34)(23,30,38,39)(35,40,36,37);(2,5)(6,7)(9,11,14,10,15,18,20,19)(12,21,24,16,25,31,33,32)(13,27,28,26,29,22,34,17)(23,36,37,39,38,35,40,30);(2,6)(4,8)(5,7)(9,12)(10,16)(11,21)(13,23)(14,24)(15,25)(17,30)(18,31)(19,32)(20,33)(22,35)(26,39)(27,36)(28,37)(29,38)(34,40);(9,13)(10,17)(11,22)(12,23)(14,28)(15,29)(16,30)(18,27)(19,26)(20,34)(21,35)(24,37)(25,38)(31,36)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,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, 255125);