#122339

order	512
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
socle	C₂²	C₂×#324
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

365 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,6,8,5)(4,7)(9,10,12,17,13,18,23,31,14,16,22,28,24,29,35,38)(11,15,20,26,21,27,34,37,19,25,32,30,33,36,40,39);(2,5,7,6)(9,11,14,19)(10,15)(12,20,22,32)(13,21,24,33)(16,25)(17,30)(18,27)(23,34,35,40)(26,28)(29,36)(31,39)(37,38);(11,19)(15,25)(20,32)(21,33)(26,30)(27,36)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=e,b^2=fh,d^2=hi,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ci,d^b=di,e^b=efhi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29535);