#130239

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	#62330	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 599 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:

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

Polycyclic presentation:

2^9|a^2=f,c^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	#72240824, #72250527, #72250535, …
maximal normal subgroup of	#72240824, #72250527, #72250535, …
maximal quotient of	#72240717

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37435);