#130295

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

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

normal subgroups only

This group has 590 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:

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

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,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	#72250025, #72250052, #72250072, …
maximal normal subgroup of	#72250025, #72250052, #72250072, …
maximal quotient of	#72250273

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37491);