#232616

order	512
nilpotency class	3
isoclinism	isoclinic class of #232606
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	C₂³×C₄	C₂⁴
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1336 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fh,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, 139812);