#144661

order	512
nilpotency class	4
isoclinism	isoclinic class of #144661
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10393
commutator, Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=d,c^2=h,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,h^b=hi,d^c=dg,e^c=ei,e^d=ei

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235749
maximal normal subgroup of	#72235749
maximal quotient of	#72235757, #72235763, #72235790

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51857);