#149018

order	512
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#11287
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

608 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,13,17,40)(4,18,22,49)(5,23,26,10)(6,27,30,11)(7,12)(8,15)(9,21)(14,41,19,50)(16,39)(20,48)(24,33)(25,36)(28,35)(29,37)(31,58,59,64)(32,45,57,44)(34,53,56,55)(38,52)(42,47)(43,62)(46,61)(51,60)(54,63);(1,3,6,16)(2,8,11,32)(4,19,21,52)(5,24,25,45)(7,17,30,46)(9,31,18,47)(10,13,36,39)(12,33,27,57)(14,28,43,22)(15,29,44,26)(20,51,53,64)(23,40,37,61)(34,62,63,50)(35,59,49,60)(38,48,41,56)(42,55,58,54);(1,4,5,20)(2,9,10,34)(3,14,15,42)(6,28,29,53)(7,22,26,54)(8,31,13,38)(11,18,37,48)(12,35,23,56)(16,43,45,64)(17,19,24,51)(21,25,55,30)(27,49,36,63)(32,60,61,41)(33,59,40,62)(39,50,57,47)(44,58,46,52)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi,e^d=ei,f^d=fi,g^d=gi,f^e=fi,g^e=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56214);