#238928

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	#60907	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 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)(3,6)(4,7)(5,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,29)(22,37)(23,32)(25,36)(27,31)(28,35)(30,34)(33,39)(38,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,33)(18,21)(19,36)(20,32)(22,30)(26,37)(28,39)(29,38)(31,35)(34,40);(2,6)(3,4)(7,8)(9,12,13,28,15,29,14,30)(10,17,19,35,20,26,18,34)(11,22,24,38,25,39,23,27)(16,31,21,33,32,40,36,37);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,29,30)(16,21,32,36)(17,34,26,35)(22,27,39,38)(31,33,40,37)

Polycyclic presentation:

2^9|c^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=ghi,f^c=fhi,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, 146124);