#240023

order	512
nilpotency class	3
isoclinism	isoclinic class of #240017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
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

863 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)(4,9)(5,7)(6,10)(8,12)(13,15)(14,16)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31);(2,7)(3,12)(4,5)(8,16)(9,10)(11,15)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,3,5,14,6,12,4,13)(2,8,9,16,10,11,7,15)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,6,5)(2,9,10,7)(3,13,12,14)(8,16,11,15)(19,26)(22,30)(27,32)(29,31)

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=dhi,e^b=eh,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, 147219);