#242369

order	512
nilpotency class	3
isoclinism	isoclinic class of #242365
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

657 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:

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

Polycyclic presentation:

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

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, 149565);