#150838

order	512
nilpotency class	5
isoclinism	isoclinic class of #150429
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×#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

373 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 64 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)(9,10,12,18,14,16,25,33)(11,20,22,35,23,30,38,39)(13,19,29,36,28,17,26,34)(15,24,31,27,32,37,40,21);(2,6)(3,4)(7,8)(9,11,14,23)(10,15,16,32)(12,24,25,37)(13,27,28,21)(17,30,19,20)(18,35,33,39)(22,26,38,29)(31,36,40,34);(1,3,5,4)(2,6,8,7)(10,16)(11,21)(12,25)(13,28)(15,30)(20,32)(22,37)(23,27)(24,38)(31,35)(34,36)(39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=cg,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,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, 58034);