#264837

order	512
nilpotency class	3
isoclinism	isoclinic class of #264819
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	#61407	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

829 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)(3,13)(4,9)(5,15)(6,11)(7,12)(8,14)(10,16)(19,27)(20,21)(23,31)(24,25)(28,32)(29,30);(1,3,6,14,7,4,16,15)(2,8,10,13,12,5,11,9)(17,18,22,26)(19,28,30,31)(20,25,21,24)(23,27,32,29);(1,4)(2,9)(3,7)(5,11)(6,15)(8,10)(12,13)(14,16)(17,19,22,30)(18,23,26,32)(20,29,21,27)(24,28,25,31);(1,5,6,9,7,8,16,13)(2,4,10,15,12,3,11,14)

Polycyclic presentation:

2^9|b^2=ei,c^2=i,d^2=e,e^2=h,f^2=i,g^2=hi,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=ghi,d^c=dh,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, 172033);