#512692

order	512
nilpotency class	5
isoclinism	isoclinic class of #33982
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#3417	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

761 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 274 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) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2,3,4)(6,12)(8,21)(9,23)(10,27)(13,31)(14,37)(15,40)(16,33)(18,44)(19,46)(20,43)(22,26)(24,53)(25,30)(28,38)(29,32)(34,57)(35,60)(36,62)(39,54)(41,56)(42,61)(45,48)(47,67)(49,58)(50,51)(52,55)(59,68)(63,65)(64,66);(5,6,11,16)(7,13,17,34)(8,22,25,52)(9,26,28,55)(10,29,27,15)(12,30,33,21)(14,38,39,23)(18,45,41,63)(19,48,42,65)(20,49,43,36)(24,54,50,37)(31,56,57,44)(32,51,40,53)(35,61,59,46)(47,68,64,60)(58,66,62,67);(5,7)(6,13)(8,18)(9,19)(10,20)(11,17)(12,31)(14,35)(15,36)(16,34)(21,44)(22,45)(23,46)(24,47)(25,41)(26,48)(27,43)(28,42)(29,49)(30,56)(32,58)(33,57)(37,60)(38,61)(39,59)(40,62)(50,64)(51,66)(52,63)(53,67)(54,68)(55,65);(7,17)(13,34)(18,41)(19,42)(20,43)(31,57)(35,59)(36,49)(44,56)(45,63)(46,61)(47,64)(48,65)(58,62)(60,68)(66,67)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419888);