#263786

order	512
nilpotency class	3
isoclinism	isoclinic class of #263779
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	#62436	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

877 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)(4,14)(5,8)(6,9)(7,12)(11,16)(13,15)(18,24)(19,26)(20,28)(21,22)(25,27)(29,30);(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,21,24,23,19,22,26)(20,25,28,31,30,27,29,32);(1,3,6,11)(2,7,9,15)(4,13,5,12)(8,16,14,10)(17,19,23,18)(20,25,30,27)(21,26,22,24)(28,31,29,32);(3,11)(7,15)(10,16)(12,13)(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=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,f^c=fi,g^c=gi

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, 170982);