#262329

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
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×#1656	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

940 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|b^2=eh,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=dhi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169525);