#147743

order	512
nilpotency class	4
isoclinism	isoclinic class of #129437
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂×#535	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1119 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1119 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=ghi,d^c=di,e^c=ei,g^c=ghi,g^d=gi,g^e=gi

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, 54939);