#449092

order	512
nilpotency class	3
isoclinism	isoclinic class of #449091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15930
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

781 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 781 conjugacy classes of subgroups — too many to draw. Only its 164 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)(3,8,15,30)(4,9)(5,20)(6,24)(7,11)(10,23)(12,27)(13,32,40,28)(14,42,21,49)(16,37,25,31)(17,45)(18,38)(19,34)(22,51)(26,35)(29,52,57,43)(33,48)(36,53)(39,61,46,64)(41,56,44,58)(47,50)(54,60)(55,62,59,63);(1,3)(2,8,11,30)(4,13)(5,21,22,14)(6,25)(7,15)(9,32,34,28)(10,29)(12,37,24,31)(16,27)(17,46,47,39)(18,44)(19,40)(20,49)(23,43,53,52)(26,41)(33,59)(35,56,38,58)(36,57)(42,51)(45,64)(48,62,54,63)(50,61)(55,60);(1,4,7,19)(2,9,11,34)(3,13,15,40)(5,17,22,47)(6,26,27,18)(8,28,30,32)(10,33,36,60)(12,38,24,35)(14,39,21,46)(16,44,25,41)(20,50,51,45)(23,54,53,48)(29,55,57,59)(31,58,37,56)(42,61,49,64)(43,63,52,62);(1,5)(2,10,11,36)(3,14)(4,17)(6,23)(7,22)(8,29,30,57)(9,33,34,60)(12,20,24,51)(13,39)(15,21)(16,43)(18,48)(19,47)(25,52)(26,54)(27,53)(28,55,32,59)(31,42,37,49)(35,45,38,50)(40,46)(41,62)(44,63)(56,64,58,61)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=i,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356288);