#254969

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
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	#59049	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1014 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,20,24)(12,17,13,21)(16,22,23,19)(25,26,28,31)(27,33,35,30)(29,32,37,40)(34,39,36,38);(2,5)(6,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,17)(18,24)(21,23)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,37)(39,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,19)(13,22)(14,20)(15,21)(17,24)(18,23)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(2,5)(6,8)(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,16,24,23)(26,31)(30,33)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=hi,c^2=ei,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 162165);