#139807

order	512
nilpotency class	4
isoclinism	isoclinic class of #139806
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9497
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3485	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 646 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)(3,13,18,42)(4,19)(5,23,14,10)(6,26,30,11)(7,12)(8,16)(9,21,35,28)(15,44,47,63)(17,49)(20,54,43,53)(22,38)(24,33)(25,37)(27,41)(29,39)(31,59,61,64)(32,48,58,51)(34,57)(36,56)(40,50)(45,52)(46,60)(55,62);(1,3,14,16,7,18,5,24)(2,8,23,42,12,33,10,13)(4,15,43,45,22,47,20,55)(6,27,29,51,30,17,25,48)(9,31,34,60,35,61,36,40)(11,32,39,49,26,58,37,41)(19,52,54,63,38,62,53,44)(21,50,57,64,28,46,56,59);(1,4,7,22)(2,9)(3,15,18,47)(5,20,14,43)(6,28,30,21)(8,31)(10,36)(11,38)(12,35)(13,40)(16,45,24,55)(17,50,27,46)(19,26)(23,34)(25,57,29,56)(32,62)(33,61)(37,54)(39,53)(41,63)(42,60)(44,49)(48,64,51,59)(52,58)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,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, 47003);