#149606

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#99
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×M₃₂	C₂³
socle	C₂	C₂×#1322
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

251 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fi,c^b=ci,e^b=efi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56802);