#237500

order	512
nilpotency class	3
isoclinism	isoclinic class of #237483
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1250 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:

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

Polycyclic presentation:

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

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, 144696);