#242370

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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=hi,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, 149566);