#124854

order	512
nilpotency class	4
isoclinism	isoclinic class of #124835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#329
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

392 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=i,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32050);