#130240

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	#62327	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

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

normal subgroups only

This group has 597 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=hi,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, #72250544, #72250559, …
maximal normal subgroup of	#72240824, #72250544, #72250559, …
maximal quotient of	#72240708

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37436);