#350472

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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1318 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)(9,10,11,14)(12,15,13,16)(18,24)(19,28)(20,29)(22,23)(25,31)(30,32);(2,5)(6,8)(10,14)(15,16)(17,18)(19,25)(20,30)(21,24)(22,26)(23,27)(28,32)(29,31);(1,3,4,7)(2,6,5,8)(10,15)(12,13)(14,16)(17,19)(18,25)(20,21)(22,29)(23,28)(24,30)(26,31)(27,32);(10,16)(12,13)(14,15)(17,20)(18,25)(19,21)(22,29)(23,28)(24,30)(26,32)(27,31)

Polycyclic presentation:

2^9|a^2=i,c^2=eh,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, 257668);