#232894

order	512
nilpotency class	3
isoclinism	isoclinic class of #232889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60951	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1270 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)(9,10,15,20)(11,21,24,35)(12,25,28,33)(13,19,14,18)(16,29,31,23)(17,27,34,26)(22,30,39,32)(36,37,40,38);(2,6)(3,4)(7,8)(9,11)(10,16)(12,22)(13,29)(14,23)(15,24)(17,32)(18,35)(19,21)(20,31)(25,36)(26,38)(27,37)(28,39)(30,34)(33,40);(1,2)(3,6)(4,7)(5,8)(9,12)(10,17)(11,22)(13,26)(14,27)(15,28)(16,30)(18,25)(19,33)(20,34)(21,36)(23,37)(24,39)(29,38)(31,32)(35,40);(1,3,5,4)(2,6,8,7)(9,13,15,14)(10,18,20,19)(11,23,24,29)(12,26,28,27)(16,21,31,35)(17,25,34,33)(22,37,39,38)(30,36,32,40)

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,e^c=ei,f^c=fi,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, 140090);