#130855

order	512
nilpotency class	4
isoclinism	isoclinic class of #130854
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	#61260	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 557 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=hi,f^2=h,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,e^c=ei,f^c=fg,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250520, #72250531, #72250559, …
maximal normal subgroup of	#72250520, #72250531, #72250559, …
maximal quotient of	#72250852

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38051);