#146688

order	512
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1142 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 190 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)(7,17,20,30)(8,21,24,28)(9,15,10,14)(11,16)(12,19)(13,23)(18,26)(22,29)(25,27)(31,34,35,32)(33,36);(1,2,3,4)(5,7)(6,12,16,27)(8,18)(9,19)(10,25)(11,20)(13,26,29,36)(14,17,15,30)(21,31,28,35)(22,32)(23,34)(24,33);(5,8,11,24)(6,13,16,29)(7,18,20,33)(9,23,10,22)(12,26,27,36)(14,21,15,28)(17,31,30,35)(19,34,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53884);