#238404

order	512
nilpotency class	3
isoclinism	isoclinic class of #238399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

796 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)(3,5)(4,11,13,16)(6,8)(7,12,9,15)(10,14)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,32)(30,31);(1,3,5,2)(4,7)(6,10,14,8)(9,13)(11,12)(15,16)(20,21)(23,31)(24,29)(27,30);(1,4)(2,7)(3,9)(5,13)(6,15)(8,16)(10,11)(12,14)(17,19)(18,23)(20,27)(21,30)(22,28)(24,32)(25,31)(26,29);(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,32,31,26)

Polycyclic presentation:

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

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, 145600);