#108539

order	512
nilpotency class	3
isoclinism	isoclinic class of #108535
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₄⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

442 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,f^b=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, 15735);