#134323

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	#1464
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

398 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 110 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)(5,6,9,15,10,14,24,30,11,16,23,31,8,13,22,29)(7,17,20,32,21,27,34,36,18,28,33,35,19,12,25,26);(1,2,3,4)(5,7)(6,12,16,27)(8,19)(9,20)(10,21)(11,18)(13,28,14,17)(15,26,31,36)(22,25)(23,33)(24,34)(29,35,30,32);(6,13,16,14)(7,18)(9,23)(12,17,27,28)(15,29,31,30)(19,21)(22,24)(26,32,36,35)

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=g,d^2=i,e^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,f^c=fgi,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, 41519);