#243998

order	512
nilpotency class	3
isoclinism	isoclinic class of #233367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224711, #72224717, #72224720, …
maximal normal subgroup of	#72224711, #72224717, #72224720, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151194);