#238026

order	512
nilpotency class	3
isoclinism	isoclinic class of #238025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

841 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145222);