#333438

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

888 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,e^b=eh,g^b=gi,g^c=gi,e^d=eh

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, 240634);