#150914

order	512
nilpotency class	5
isoclinism	isoclinic class of #150425
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
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

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

normal subgroups only

This group has 318 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=gi,d^2=ghi,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfh,d^b=dg,e^b=efgh,d^c=dhi,f^c=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58110);