#350308

order	512
nilpotency class	3
isoclinism	isoclinic class of #346294
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×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1202 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)(17,18,23,27)(19,24,22,26)(20,29,25,32)(21,28,31,30);(2,5)(6,8)(10,14)(15,16)(17,19,23,22)(18,24,27,26)(20,28,25,30)(21,29,31,32);(1,3,4,7)(2,6,5,8)(10,15)(12,13)(14,16)(17,20,23,25)(18,21,27,31)(19,28,22,30)(24,29,26,32);(10,16)(12,13)(14,15)(17,21)(18,25)(19,29)(20,27)(22,32)(23,31)(24,30)(26,28)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,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, 257504);