#487208

order	512
nilpotency class	4
isoclinism	isoclinic class of #486819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

716 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394404);