#480053

order	512
nilpotency class	4
isoclinism	isoclinic class of #33348
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#99
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂²×#324
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

951 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 951 conjugacy classes of subgroups — too many to draw. Only its 312 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,g^a=gi,e^b=ei,f^b=fg,d^c=di,f^e=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387249);