#99150

order	512
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

732 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,7,8,5)(4,6)(9,10,12,15,13,16,11,14)(17,18)(19,25,27,22)(20,28)(21,24)(23,29)(26,30,32,31);(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(1,3,4,8)(2,5,6,7)(17,20)(18,23,24,28)(19,26)(21,29)(22,30,25,31)(27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=dh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,e^c=eg,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346);