#130853

order	512
nilpotency class	4
isoclinism	isoclinic class of #130850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8376
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

547 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,13,5,10,27,37,8,14,29,45,7,12,31,42)(3,15,18,47,17,46,52,64,20,48,53,60,19,38,28,58)(4,21,25,44,24,39,56,63,23,11,30,40,26,36,57,62)(9,32,34,49,41,54,43,22,35,55,61,16,33,51,59,50);(1,3,8,20)(2,9,14,35)(4,22,23,50)(5,19,7,17)(6,28,29,52)(10,33,12,41)(11,38,21,46)(13,43,45,59)(15,36,48,39)(16,24,49,26)(18,27,53,31)(25,54,30,51)(32,57,55,56)(34,42,61,37)(40,47,44,60)(58,63,64,62);(2,10,14,12)(4,23)(6,29)(9,33,35,41)(11,39,21,36)(13,37,45,42)(15,46,48,38)(16,49)(18,53)(22,50)(24,26)(27,31)(28,52)(34,59,61,43)(40,63,44,62)(47,64,60,58)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250531, #72250549, #72250762, …
maximal normal subgroup of	#72250531, #72250549, #72250762, …
maximal quotient of	#72250850

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38049);