#242392

order	512
nilpotency class	3
isoclinism	isoclinic class of #242389
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

585 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=dh,g^b=gi,e^c=eh,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149588);