#263917

order	512
nilpotency class	3
isoclinism	isoclinic class of #263899
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

897 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)(3,10)(4,15)(5,8)(6,9)(7,12)(11,13)(14,16)(17,18)(19,27)(20,22)(21,25)(23,26)(24,29)(28,31)(30,32);(1,2,5,15,6,9,4,8)(3,11,13,10,14,7,12,16)(17,18,22,25,23,26,21,20)(19,28,31,27,30,24,29,32);(1,2)(3,7)(4,8)(5,15)(6,9)(10,12)(11,14)(13,16)(17,19)(18,24)(20,27)(21,29)(22,31)(23,30)(25,32)(26,28);(1,3,4,12,6,14,5,13)(2,7,8,10,9,11,15,16)(17,20,22,18,23,25,21,26)(19,27,31,24,30,32,29,28)

Polycyclic presentation:

2^9|b^2=eh,d^2=ei,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,d^b=dh,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171113);