#350045

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
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	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

881 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)(10,13)(11,15)(14,16)(18,23)(19,28)(21,22)(24,31)(25,32)(29,30);(2,5)(6,8)(17,18)(19,25)(20,23)(21,26)(22,27)(24,29)(28,31)(30,32);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,19,20,29)(18,24,23,25)(21,30,22,28)(26,31,27,32);(9,10)(11,16)(12,14)(13,15)(17,19,20,29)(18,25,23,24)(21,28,22,30)(26,31,27,32)

Polycyclic presentation:

2^9|c^2=e,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,c^b=ch,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257241);