#109670

order	512
nilpotency class	3
isoclinism	isoclinic class of #9385
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

868 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 170 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,d^b=di,e^b=ei,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16866);