#6438856

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3037 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 68 points:

(2,3)(5,6,12,17)(7,13,21,38)(8,22,27,41)(9,15,30,46)(10,31,11,16)(14,25,42,26)(18,47,50,39)(19,35,52,60)(20,37,32,36)(23,34,49,59)(24,53,57,63)(28,43,51,61)(29,45,33,44)(40,55,64,56)(48,65,67,62)(54,58,66,68);(1,2,4,3)(5,7)(6,13)(8,23)(9,28)(10,20)(11,32)(12,21)(14,39)(15,43)(16,36)(17,38)(18,25)(19,29)(22,34)(24,48)(26,50)(27,49)(30,51)(31,37)(33,52)(35,45)(40,58)(41,59)(42,47)(44,60)(46,61)(53,65)(54,56)(55,66)(57,67)(62,63)(64,68);(1,3,4,2)(5,8)(6,14)(7,18)(9,24)(10,25)(11,26)(12,27)(13,34)(15,40)(16,41)(17,42)(19,48)(20,49)(21,50)(22,31)(23,32)(28,54)(29,55)(30,57)(33,56)(35,58)(36,39)(37,47)(38,59)(43,62)(44,63)(45,53)(46,64)(51,66)(52,67)(60,68)(61,65);(5,9)(6,15)(7,19)(8,24)(10,29)(11,33)(12,30)(13,35)(14,40)(16,44)(17,46)(18,48)(20,51)(21,52)(22,53)(23,54)(25,55)(26,56)(27,57)(28,32)(31,45)(34,58)(36,43)(37,61)(38,60)(39,62)(41,63)(42,64)(47,65)(49,66)(50,67)(59,68);(5,10,12,11)(6,16,17,31)(7,20,21,32)(8,25,27,26)(9,29,30,33)(13,36,38,37)(14,41,42,22)(15,44,46,45)(18,49,50,23)(19,51,52,28)(24,55,57,56)(34,39,59,47)(35,43,60,61)(40,63,64,53)(48,66,67,54)(58,62,68,65)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346052);