#290258

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=efi,c^2=i,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

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, 197454);