#124850

order	512
nilpotency class	4
isoclinism	isoclinic class of #124850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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	C₂²×D₄	M₁₆
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

298 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 70 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)(9,10,12,18,13,16,24,31,14,17,23,32,25,30,36,40)(11,19,21,33,22,27,35,38,20,15,26,28,34,29,37,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,29)(17,19)(18,33)(23,26)(24,35)(25,34)(27,30)(28,32)(31,38)(36,37)(39,40);(10,16,17,30)(11,20)(15,27,19,29)(18,31,32,40)(21,26)(22,34)(28,38,33,39)(35,37)

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,c^b=ci,d^b=dg,e^b=efg,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, 32046);