#263263

order	512
nilpotency class	3
isoclinism	isoclinic class of #263259
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3506	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

859 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)(3,11)(4,7)(5,14)(6,10)(8,12)(9,15)(13,16)(17,18)(19,25)(20,28)(21,24)(22,26)(23,29)(27,32)(30,31);(1,3,5,4,6,12,15,13)(2,7,9,11,10,16,14,8)(20,29)(22,31)(23,28)(26,30);(17,19,21,27)(18,22,24,31)(20,30,29,26)(23,32,28,25);(1,4,5,12,6,13,15,3)(2,8,9,7,10,11,14,16)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,f^c=fi

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, 170459);