#235633

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

1269 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)(20,25)(21,23)(22,26)(24,29)(28,32)(30,31);(1,2)(3,11)(4,10)(5,7)(6,9)(8,13)(12,15)(14,16)(18,23)(19,28)(20,21)(25,26)(27,31)(29,30);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,16)(10,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(3,12)(8,16)(11,15)(13,14)(17,20,22,21)(18,25,26,23)(19,28,30,29)(24,27,32,31)

Polycyclic presentation:

2^9|d^2=i,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, 142829);