#255027

order	512
nilpotency class	3
isoclinism	isoclinic class of #246503
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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

978 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)(3,6)(4,5)(7,8)(9,10,14,11)(12,16,13,15)(17,18)(19,29)(20,24)(21,28)(22,27)(23,26)(25,31)(30,32);(2,5)(6,8)(9,11,14,10)(12,15,13,16)(17,19,22,20)(18,24,27,29)(21,30,23,31)(25,26,32,28);(1,3,4,7)(2,6,5,8)(10,15)(11,16)(12,13)(17,20,22,19)(18,25,27,32)(21,31,23,30)(24,28,29,26);(2,5)(6,8)(9,12,14,13)(10,16,11,15)(17,21,22,23)(18,26,27,28)(19,30,20,31)(24,32,29,25)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 162223);