#483010

order	512
nilpotency class	4
isoclinism	isoclinic class of #483009
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#329
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#97	C₂⁴
prime radical	#3499	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ #2875 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

784 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 232 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=h,f^2=g,g^2=i,b^a=be,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,f^e=fi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390206);