#144403

order	512
nilpotency class	4
isoclinism	isoclinic class of #144403
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10337
commutator, Frattini, cosocle	C₂²×#41	C₂³
prime radical	#23898	C₂
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

725 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 725 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285029, #72286146
maximal normal subgroup of	#72285029, #72286146

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51599);