#137971

order	512
nilpotency class	4
isoclinism	isoclinic class of #125110
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#32939	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

604 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=gh,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45167);