#237181

order	512
nilpotency class	3
isoclinism	isoclinic class of #237155
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

1178 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144377);