#240861

order	512
nilpotency class	3
isoclinism	isoclinic class of #10508
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	#60790	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

984 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)(4,7)(6,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,29)(22,37)(23,32)(25,33)(27,40)(28,36)(30,35)(31,38)(34,39);(3,8)(5,6)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,21)(19,33)(20,32)(22,30)(26,37)(28,39)(29,38)(31,35)(36,40);(2,5)(3,8)(6,7)(9,12,13,28,15,29,14,30)(10,17,19,36,20,26,18,35)(11,22,24,38,25,39,23,27)(16,31,21,37,32,40,33,34);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,29,30)(16,21,32,33)(17,35,26,36)(22,27,39,38)(31,37,40,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148057);