#134271

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1467
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

388 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=g,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ci,e^c=ei,f^c=fg,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, 41467);