#108384

order	512
nilpotency class	3
isoclinism	isoclinic class of #108372
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

876 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

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

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, 15580);