#262255

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

839 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)(7,11)(8,12)(9,13)(10,14)(15,19)(16,18)(17,20)(21,22,26,24)(23,27,30,32)(25,33,35,28)(29,31,36,34);(1,2,4,3)(5,7)(6,11)(8,16)(9,15)(10,17)(12,18)(13,19)(14,20)(21,23)(22,27)(24,32)(25,34)(26,30)(28,36)(29,33)(31,35);(6,12)(7,15)(8,9)(13,14)(16,17)(18,19)(21,24,26,22)(23,29,30,36)(25,28,35,33)(27,34,32,31);(1,3,4,2)(5,8,10,9)(6,12,14,13)(7,16,17,15)(11,18,20,19)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=eh,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, 169451);