#233858

order	512
nilpotency class	3
isoclinism	isoclinic class of #233856
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
prime radical	#60798	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1412 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:

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

Polycyclic presentation:

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

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, 141054);