#274746

order	512
nilpotency class	3
isoclinism	isoclinic class of #267131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181942);