#150493

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	#391	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

367 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 58 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)(9,10,13,18,15,20,12,26)(11,21,24,16,25,33,39,35)(14,19,28,36,31,38,29,17)(22,37,27,32,30,40,23,34);(2,6)(3,4)(7,8)(9,11,15,25)(10,16,20,35)(12,27,13,23)(14,30,31,22)(17,21,36,33)(18,37,26,40)(19,32,38,34)(24,28,39,29);(1,3,5,4)(2,7,8,6)(11,22)(14,31)(16,32)(17,36)(19,38)(21,37)(23,39)(24,27)(25,30)(28,29)(33,40)(34,35)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cfh,d^b=dg,e^b=efgi,f^b=fi,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, 57689);