#339963

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

616 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,f^c=fi,f^d=fi

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, 247159);