#480019

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
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	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂²×#323
prime radical	C₂×#585	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1503 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 1503 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:

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

Polycyclic presentation:

2^9|a^2=f,e^2=i,f^2=h,h^2=i,b^a=be,e^a=eg,g^a=gi,e^b=ei,f^b=fgi,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, 387215);