#151998

order	512
nilpotency class	5
isoclinism	isoclinic class of #1426
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#367
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	C₄×#367
prime radical	C₂³	#508
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

197 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 112 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=degh,d^2=ghi,e^2=ghi,g^2=hi,h^2=i,b^a=bd,f^a=fi,d^b=dg,e^b=egi,g^b=gh,h^b=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223679, C₂×#151998, #72225313, …
maximal normal subgroup of	#72223679, C₂×#151998, #72225313, …
maximal quotient of	C₈×#1426, C₂×#151998

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59194);