#130297

order	512
nilpotency class	4
isoclinism	isoclinic class of #130294
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	#62335	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

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

normal subgroups only

This group has 588 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:

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

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=ei,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,e^d=ei,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250056, #72250072, #72250076, …
maximal normal subgroup of	#72250056, #72250072, #72250076, …
maximal quotient of	#72250564, #72280423

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37493);