#147803

order	512
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2021
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

1028 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1028 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)(3,14)(4,19,24,41)(5,25,27,11)(6,12)(7,29)(8,13)(9,16,33,51)(10,22,36,23)(15,46,48,64)(17,43)(18,45)(20,54,52,42)(21,56)(26,39,28,38)(30,44)(31,60)(32,49,40,50)(34,63)(35,59)(37,57)(47,62)(53,58)(55,61);(1,3)(2,9)(4,20)(5,16)(6,17)(7,30)(8,18)(10,34)(11,14)(12,40)(13,33)(15,23)(19,31)(21,55)(22,48)(24,52)(25,45)(26,49)(27,51)(28,50)(29,32)(35,46)(36,62)(37,64)(38,44)(39,43)(41,61)(42,56)(47,57)(53,54)(58,60)(59,63);(1,4)(2,10)(3,15)(5,21)(6,23)(7,22)(8,24)(9,31)(11,37)(12,41)(13,36)(14,42)(16,47)(17,52)(18,48)(19,29)(20,30)(25,35)(26,59)(27,58)(28,57)(32,62)(33,61)(34,40)(38,56)(39,53)(43,64)(44,46)(45,54)(49,55)(50,60)(51,63)

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,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, 54999);