#235361

order	512
nilpotency class	3
isoclinism	isoclinic class of #235354
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1273 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,d^c=dhi,e^c=ei,f^c=fhi

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, 142557);