#150841

order	512
nilpotency class	5
isoclinism	isoclinic class of #150425
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	#2011	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

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

normal subgroups only

This group has 389 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,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=cfhi,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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58037);