#290229

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

545 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)(9,10,14,16,12,15,13,11)(17,18,20,30,22,25,21,24)(19,23,27,32,29,26,28,31);(1,3,5,4)(2,6,8,7)(9,11,14,10,12,16,13,15)(17,19,21,28,22,29,20,27)(18,23,24,31,25,26,30,32);(9,10,12,15)(11,13,16,14)(17,18,22,25)(19,26,29,23)(20,24,21,30)(27,32,28,31);(1,4,5,3)(2,7,8,6)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,26,32)

Polycyclic presentation:

2^9|a^2=efi,b^2=ef,c^2=h,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, 197425);