#139756

order	512
nilpotency class	4
isoclinism	isoclinic class of #139755
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9495
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3485	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

671 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,13)(4,18)(5,22,24,10)(6,25,27,11)(7,12)(8,15,30,44)(9,20,32,51)(14,38,41,63)(16,37,26,56)(17,33)(19,46,49,61)(21,47)(23,35)(28,57,58,64)(29,53)(31,59)(34,48)(36,40)(39,45)(42,62)(43,55)(50,60)(52,54);(1,3,5,15)(2,8,10,33)(4,14,19,42)(6,26,54,43)(7,17,24,44)(9,28,34,60)(11,29,52,37)(12,30,22,13)(16,23,53,27)(18,45,46,38)(20,50,59,57)(21,41,49,39)(25,55,35,56)(31,36,32,58)(40,48,64,51)(47,62,61,63);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,28)(10,34)(11,18)(12,32)(13,36)(15,42)(16,40)(17,41)(22,31)(23,48)(24,49)(25,47)(26,50)(27,51)(29,45)(30,58)(33,60)(35,61)(37,38)(39,44)(43,57)(46,52)(53,64)(54,59)(55,62)(56,63)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,f^d=fi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209847, #72258402
maximal normal subgroup of	#72209847, #72258402
maximal quotient of	#72227683, #72227719, #72227723, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46952);