#140581

order	512
nilpotency class	4
isoclinism	isoclinic class of #13373
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#486
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9644
prime radical	#3150	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 385 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283857, #72283870, #72283926, …
maximal normal subgroup of	#72283857, #72283870, #72283926, …
maximal quotient of	#72210536, #72210557, #72210567, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47777);