#147795

order	512
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10995
prime radical	#60749	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1042 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1042 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,13,17,37)(4,18,22,9)(5,23,25,10)(6,27)(7,11)(8,15)(12,30)(14,44)(16,48,28,43)(19,51)(20,35)(21,36,29,41)(24,33)(26,39,59,56)(31,57,61,45)(32,58)(34,52,50,53)(38,54)(40,47)(42,46)(49,62)(55,64)(60,63);(1,3)(2,8)(4,19)(5,24)(6,28)(7,17)(9,34)(10,37)(11,33)(12,40)(13,23)(14,21)(15,25)(16,30)(18,50)(20,53)(22,49)(26,47)(27,32)(29,46)(31,36)(35,51)(38,62)(39,43)(41,61)(42,63)(44,64)(45,60)(48,56)(52,54)(55,57)(58,59);(1,4)(2,9)(3,14)(5,20)(6,29)(7,22)(8,31)(10,38)(11,18)(12,41)(13,42)(15,45)(16,49)(17,46)(19,28)(21,30)(23,35)(24,57)(25,54)(26,60)(27,36)(32,50)(33,61)(34,40)(37,44)(39,63)(43,62)(47,53)(48,51)(52,58)(55,59)(56,64)

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,g^c=gi,e^d=ei,g^d=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, 54991);