#130238

order	512
nilpotency class	4
isoclinism	isoclinic class of #130237
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

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

normal subgroups only

This group has 587 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dgi,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	#72307312, #72307364, #72307553, …
maximal normal subgroup of	#72307312, #72307364, #72307553, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37434);