#130235

order	512
nilpotency class	4
isoclinism	isoclinic class of #130232
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8350
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 585 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307081, #72307106, #72307112, …
maximal normal subgroup of	#72307081, #72307106, #72307112, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37431);