#149526

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×#1314
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 365 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fi,c^b=ci,d^b=di,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, 56722);