#263871

order	512
nilpotency class	3
isoclinism	isoclinic class of #263859
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	#3498	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

905 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)(18,23)(19,26)(20,29)(21,25)(27,28)(31,32);(1,3,6,14,7,4,16,15)(2,8,10,13,12,5,11,9)(17,18,21,23,22,20,25,29)(19,24,27,31,28,30,26,32);(1,4)(2,9)(3,7)(5,11)(6,15)(8,10)(12,13)(14,16)(17,19,22,28)(18,24,20,30)(21,27,25,26)(23,31,29,32);(1,5,6,9,7,8,16,13)(2,4,10,15,12,3,11,14)(17,20,21,29,22,18,25,23)(19,24,27,31,28,30,26,32)

Polycyclic presentation:

2^9|b^2=e,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,c^b=cg,g^b=ghi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171067);