#130792

order	512
nilpotency class	4
isoclinism	isoclinic class of #130786
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8375
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#61090	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

805 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 805 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

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,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=fgi,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250043, #72250056, #72250223, …
maximal normal subgroup of	#72250043, #72250056, #72250223, …
maximal quotient of	#72250434, #72280388

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37988);