#144220

order	512
nilpotency class	4
isoclinism	isoclinic class of #144219
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#32942	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 576 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)(3,5)(4,6)(7,8)(9,10,15,21)(11,22,25,36)(12,16,29,33)(13,20,14,19)(17,30,37,23)(18,28,32,26)(24,34,27,35)(31,39,38,40);(1,3)(2,5)(4,7)(6,8)(10,16)(11,23)(12,26)(13,14)(17,34)(18,20)(19,32)(21,33)(22,38)(24,27)(25,30)(28,29)(31,36)(35,37);(5,8)(9,11)(10,17)(12,27)(13,30)(14,23)(15,25)(16,31)(18,34)(19,36)(20,22)(21,37)(24,29)(26,40)(28,39)(32,35)(33,38)

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51416);