#263957

order	512
nilpotency class	3
isoclinism	isoclinic class of #263931
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	#62535	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

866 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)(20,21)(22,26)(23,25)(24,30)(28,31)(29,32);(1,2,5,15,6,9,4,8)(3,11,13,10,14,7,12,16)(17,18,23,20,22,26,21,25)(19,24,30,27,31,32,29,28);(1,2)(3,11)(4,8)(5,15)(6,9)(7,14)(10,13)(12,16)(17,19,22,31)(18,24,26,32)(20,27,25,28)(21,29,23,30);(1,3,4,12,6,14,5,13)(2,7,8,10,9,11,15,16)(17,20,23,26,22,25,21,18)(19,28,30,24,31,27,29,32)

Polycyclic presentation:

2^9|b^2=eh,c^2=i,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,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, 171153);