#119910

order	512
nilpotency class	3
isoclinism	isoclinic class of #119905
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

663 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 663 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27106);