#124837

order	512
nilpotency class	4
isoclinism	isoclinic class of #124835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	C₂×#532	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 506 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)(9,10,13,18,15,19,27,36,14,20,28,35,12,17,25,33)(11,16,23,30,21,29,37,34,24,32,38,40,22,31,26,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,26)(14,24)(15,22)(17,29)(18,34)(19,31)(20,32)(23,25)(27,38)(28,37)(30,36)(33,40)(35,39);(2,6)(5,7)(9,12,14,15)(10,17,20,19)(11,21,24,22)(13,25,28,27)(16,29,32,31)(18,33,35,36)(23,37,38,26)(30,34,40,39)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=g,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=efgi,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, 32033);