#149560

order	512
nilpotency class	5
isoclinism	isoclinic class of #149559
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11426
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	#381	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #2310 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

408 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=i,e^2=g,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=di,e^b=efi,g^b=gi,d^c=di,e^d=eh,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56756);