#257968

order	512
nilpotency class	3
isoclinism	isoclinic class of #247367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 165164);