#239282

order	512
nilpotency class	3
isoclinism	isoclinic class of #238006
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	#60688	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

892 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,23)(13,27)(14,17)(15,21)(16,26)(18,29)(20,34)(22,33)(24,38)(25,32)(28,31)(30,35)(36,39)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,24)(13,28)(14,22)(16,30)(17,31)(18,33)(19,27)(20,26)(21,29)(23,36)(25,35)(32,38)(34,37)(39,40);(2,6)(3,4)(7,8)(9,12,13,25,14,26,29,40)(10,16,18,32,17,23,27,37)(11,20,21,35,22,24,28,39)(15,30,19,34,31,36,33,38);(1,3,5,4)(2,7,8,6)(10,17)(15,31)(16,23)(18,27)(19,33)(30,36)(32,37)(34,38)

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fhi,g^b=gi,d^c=di,f^c=fhi,g^c=ghi

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, 146478);