#150852

order	512
nilpotency class	5
isoclinism	isoclinic class of #150441
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

295 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 62 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)(4,6)(5,8)(9,10,12,17,14,19,26,35)(11,20,23,38,22,37,39,34)(13,18,29,36,28,16,27,33)(15,30,31,24,32,25,40,21);(2,6)(3,4)(7,8)(9,11,14,22)(10,15)(12,25,26,30)(13,21,28,24)(16,20)(17,34)(18,37)(19,32)(23,27,39,29)(31,33)(35,38)(36,40);(1,3,5,4)(2,6,8,7)(11,21)(12,26)(13,28)(15,20)(16,18)(17,35)(22,24)(23,25)(30,39)(31,34)(32,37)(38,40)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=gh,d^2=gi,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cfhi,d^b=dg,e^b=efg,f^b=fi,d^c=dhi,e^c=ei,f^c=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58048);