#124905

order	512
nilpotency class	4
isoclinism	isoclinic class of #124849
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	#1467
prime radical	C₂³	#407
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

263 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 76 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)(9,10,13,19,14,18,28,34,15,20,27,35,12,16,26,29)(11,21,23,36,24,17,30,32,25,31,38,39,22,33,37,40);(1,2)(3,7)(4,6)(5,8)(10,16,20,18)(13,27)(17,21,33,31)(19,29,35,34)(23,38)(26,28)(30,37)(32,36,40,39);(9,11,14,24,15,25,12,22)(10,17,16,21,20,33,18,31)(13,23,28,30,27,38,26,37)(19,32,29,36,35,40,34,39)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=fh,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,d^b=dg,e^b=efgi,e^d=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, 32101);