#235353

order	512
nilpotency class	3
isoclinism	isoclinic class of #235342
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	#60959	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1291 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)(3,11)(4,13)(5,8)(6,9)(7,10)(12,15)(14,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,31)(28,32);(1,3,7,14)(2,4,10,15)(5,12,6,13)(8,11,9,16)(23,31)(26,32);(1,4)(2,3)(5,11)(6,16)(7,15)(8,12)(9,13)(10,14)(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,31)(26,30);(1,5,7,6)(2,8,10,9)(3,12,14,13)(4,11,15,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,26,31,32)

Polycyclic presentation:

2^9|b^2=h,d^2=hi,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=di,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, 142549);