#150704

order	512
nilpotency class	5
isoclinism	isoclinic class of #150594
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
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 ◁ C₂³ ◁ C₂×C₄² ◁ #2326 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

264 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

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=ch,d^b=dg,e^b=efg,f^b=fi,d^c=di,e^c=ei,e^d=ehi,f^e=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57900);