#237169

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

1184 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)(4,7)(5,9)(6,10)(8,12)(13,16)(14,15)(17,18,23,27)(19,28,30,31)(20,24,32,29)(21,25,22,26);(2,7)(3,12)(4,5)(9,10)(11,15)(13,14)(17,19)(18,20)(21,24)(22,29)(23,30)(25,28)(26,31)(27,32);(1,3,6,14)(2,8,10,16)(4,12,5,13)(7,15,9,11)(17,20,23,32)(18,24,27,29)(19,26,30,25)(21,31,22,28);(1,4,6,5)(2,7,10,9)(3,13,14,12)(8,11,16,15)(17,21,23,22)(18,25,27,26)(19,24,30,29)(20,28,32,31)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,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, 144365);