#135188

order	512
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1817
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

420 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

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, 42384);