#150849

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#390	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 301 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)(11,20,23,32)(12,16,25,35)(13,18,29,17)(14,19)(15,22,31,27)(21,33,24,30)(26,36)(28,34)(37,39,38,40);(2,6)(3,4)(7,8)(9,11,14,22)(10,15)(12,24,25,38)(13,27,29,23)(16,33)(17,20)(18,31)(19,32)(21,28,37,26)(30,34)(35,39)(36,40);(1,3,5,4)(2,6,8,7)(9,12,14,25)(10,16,19,35)(11,21,22,37)(13,28,29,26)(15,30,32,40)(17,36,18,34)(20,33,31,39)(23,38,27,24)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=ghi,d^2=gi,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, 58045);