#139841

order	512
nilpotency class	4
isoclinism	isoclinic class of #139840
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9499
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#3486	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

507 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,f^d=fi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47037);