#128406

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂×C₄²
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

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

normal subgroups only

This group has 396 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,d^a=di,f^a=fgi,c^b=ch,d^b=dg,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281057, #72281248, #72313762
maximal normal subgroup of	#72281057, #72281248, #72313762

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35602);