#241253

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 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:

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

Polycyclic presentation:

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

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, 148449);