#285634

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

580 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)(9,10,14,16,13,17,12,23)(11,18,22,19,21,15,20,24)(25,26,29,33,28,32,37,31)(27,34,36,40,35,30,38,39);(2,5)(6,8)(9,11,14,22,13,21,12,20)(10,15,16,24,17,18,23,19)(25,27,29,36,28,35,37,38)(26,30,33,39,32,34,31,40);(1,3,4,7)(2,6,5,8)(9,10)(11,19)(12,16)(13,17)(14,23)(15,20)(18,22)(21,24)(26,31)(27,35)(29,37)(30,39)(32,33)(34,40);(2,5)(6,8)(9,12,13,14)(10,16,17,23)(11,20,21,22)(15,24,18,19)(26,32)(30,34)(31,33)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192830);