#508497

order	512
nilpotency class	4
isoclinism	isoclinic class of #508496
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29892
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

593 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 593 conjugacy classes of subgroups — too many to draw. Only its 128 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)(3,14)(4,20)(5,26)(6,28)(7,11)(8,13)(9,16)(10,22)(12,31)(15,46)(17,34)(18,35)(19,39)(21,54)(23,58)(24,38)(25,40)(27,41)(29,42)(30,37)(32,60)(33,49)(36,56)(43,59)(44,48)(45,50)(47,63)(51,61)(52,64)(53,55)(57,62);(1,3)(2,9,13,14)(4,21)(5,19)(6,29)(7,17)(8,16)(10,36,38,64)(11,39,26,34)(12,42,28,61)(15,23)(18,32)(20,53,40,54)(22,52)(24,56)(25,55)(27,50)(30,48)(31,51)(33,62,44,37)(35,60,45,41)(43,46,58,63)(47,59)(49,57);(1,4,8,25)(2,10,13,38)(3,15,16,47)(5,24,7,22)(6,30,31,57)(9,33,14,44)(11,40,26,20)(12,43,28,58)(17,49,19,48)(18,52,50,56)(21,51,55,29)(23,27,59,32)(34,46,39,63)(35,54,45,53)(36,61,64,42)(37,60,62,41);(6,31)(12,28)(15,47)(18,50)(23,59)(27,32)(29,51)(30,57)(33,44)(35,45)(37,62)(41,60)(42,61)(43,58)(46,63)(48,49)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415693);