#263443

order	512
nilpotency class	3
isoclinism	isoclinic class of #263431
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	#62332	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

825 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)(3,9)(4,12)(5,8)(6,10)(7,13)(11,16)(14,15)(18,24)(19,26)(20,28)(21,22)(25,27)(29,30);(4,13)(6,15)(7,12)(10,14)(17,18,21,24,23,19,22,26)(20,25,28,31,30,27,29,32);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(17,19,23,18)(20,25,30,27)(21,26,22,24)(28,31,29,32);(17,20,22,29,23,30,21,28)(18,25,26,32,19,27,24,31)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,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, 170639);