#150819

order	512
nilpotency class	5
isoclinism	isoclinic class of #150397
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×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

318 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ghi,d^2=g,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,d^c=dh,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, 58015);