#128405

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
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	#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

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

normal subgroups only

This group has 412 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)(7,16)(8,21)(9,23)(10,14)(11,13)(12,18)(15,24)(17,28)(19,31)(20,26)(22,29)(25,30)(27,33)(32,35)(34,36);(5,7,9,19,10,18,25,33,11,20,24,34,8,17,22,32)(6,12,15,27,14,26,29,36,13,28,23,35,21,16,30,31);(1,3,4,2)(6,13)(7,17,20,18)(9,24)(12,26,28,16)(14,21)(19,33,34,32)(22,25)(27,31,35,36)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ci,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35601);