#124707

order	512
nilpotency class	4
isoclinism	isoclinic class of #124679
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	#1464
prime radical	C₂³	#407
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

247 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

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

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, 31903);