#235627

order	512
nilpotency class	3
isoclinism	isoclinic class of #235627
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1272 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142823);