#124785

order	512
nilpotency class	4
isoclinism	isoclinic class of #124660
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1510
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7620
prime radical	#3050	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,13,16,40,18,41,17,34)(4,9)(5,22,26,10)(6,12)(7,11)(8,15,29,23,30,48,38,49)(14,42,44,63,46,39,45,59)(19,35,24,31)(20,33)(21,32)(25,36)(27,54,51,37)(28,43,57,50,58,47,62,55)(52,60)(53,64,56,61);(1,3)(2,8)(4,14)(5,23)(6,16)(7,18)(9,28)(10,34)(11,30)(12,38)(13,37)(15,27)(17,25)(19,50)(20,44)(21,46)(22,40)(24,55)(26,49)(29,36)(31,59)(32,58)(33,62)(35,63)(39,64)(41,54)(42,61)(43,53)(45,52)(47,56)(48,51)(57,60);(1,4,6,20,7,21,25,52)(2,9,12,33,11,32,36,60)(3,14,16,44,18,46,17,45)(5,24,27,56,26,19,51,53)(8,28,38,62,30,58,29,57)(10,35,37,64,22,31,54,61)(13,39,40,59,41,42,34,63)(15,47,49,50,48,43,23,55)

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31981);