#146580

order	512
nilpotency class	4
isoclinism	isoclinic class of #146575
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10735
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1661 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1661 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,f^d=fi,g^d=gi,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53776);