#151999

order	512
nilpotency class	5
isoclinism	isoclinic class of #13657
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2177
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12674
prime radical	C₂³	#510
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

189 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=deg,d^2=ghi,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ch,d^a=di,f^a=fi,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223659, #72223679, C₂×#151999, …
maximal normal subgroup of	#72223659, #72223679, C₂×#151999, …
maximal quotient of	#72223657, #72223674, C₂×#151999

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59195);