#350445

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1096 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)(11,14)(15,16)(19,28)(20,21)(23,31)(24,32)(25,26)(29,30);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,29)(28,31)(30,32);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,27)(25,32)(26,31);(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,27)(25,32)(26,31)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,d^b=dhi,f^c=fh,g^c=gi,f^d=fi,g^d=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257641);