#153074

order	512
nilpotency class	5
isoclinism	isoclinic class of #149931
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2328 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 329 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eh,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dghi,e^b=ei,f^b=fgh,g^b=ghi,h^b=hi,d^c=di,f^c=fi,f^d=fhi,g^f=gi

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, 60270);