#268782

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1037 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,15,11,16,9,14)(7,12,17,20,18,13,8,19)(21,22,24,28,25,29,33,27)(23,26,30,35,31,34,32,36);(2,4)(5,7)(6,12)(8,9)(10,17)(11,18)(13,16)(14,19)(15,20)(21,23,25,31)(22,26,29,34)(24,30,33,32)(27,36,28,35);(5,8)(6,13)(7,9)(10,18)(11,17)(12,16)(14,19)(15,20)(22,27)(23,30)(24,33)(28,29)(31,32)(35,36);(2,4)(5,9,11,10)(6,14,16,15)(7,17,18,8)(12,20,13,19)(23,31)(26,34)(30,32)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175978);