#262289

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#562	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

855 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169485);