#240192

order	512
nilpotency class	3
isoclinism	isoclinic class of #240183
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60559	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

940 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)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,30)(22,37)(23,32)(25,36)(27,31)(28,35)(29,34)(33,39)(38,40);(9,11,15,25)(10,16,20,32)(12,27,30,38)(13,23,14,24)(17,33,26,37)(18,21,19,36)(22,29,39,28)(31,35,40,34);(2,6)(3,4)(7,8)(9,12,13,29,15,30,14,28)(10,17,19,34,20,26,18,35)(11,22,24,38,25,39,23,27)(16,31,21,33,32,40,36,37);(1,3,5,4)(2,7,8,6)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,21,32,36)(17,34,26,35)(22,38,39,27)(31,37,40,33)

Polycyclic presentation:

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

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, 147388);